MEROMORPHIC INNER FUNCTIONS, TOEPLITZ KERNELS, AND THE UNCERTAINTY PRINCIPLE

N. MAKAROV AND A. POLTORATSKI

Introduction

This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation to the theory of complex Fourier transform of compactly supported measures or, more generally, Fourier-Weyl-Titchmarsh transforms associated with selfadjoint differential operators with compact resolvent.

The last part of the title is a reference to the monograph [19], which contains a large collection of results that could be described by the (informal) statement: ”it is impossible for a non-zero function and its Fourier transform to be simultaneously very small.” For example, if a function is supported on a small interval, then the set of zeros of its Fourier transform has to be sparse. Another example: a small amount of information about the potential of a Schrödinger operator requires a large amount of information about the spectral measure to determine the operator uniquely.

Our goal is to present a unified approach to certain problems of this type. The method is based on the reduction (complete or partial) to the injectivity problem for Toeplitz operators, which makes it possible to use the full strength of the theory of Hardy-Nevanlinna classes and Hilbert transform rather than rely on the standard techniques of the theory of entire function (Jensen type formulae, canonical products, Phragmen-Lindelöf principle). We have been able to reinterpret various classical results (often with shorter proofs and stronger conclusions) in a way that allows further generalizations. We explain the reason for such generalizations (and their nature) below. We begin by briefly describing the background.

0.1. Complex exponentials, Paley-Wiener spaces, and Cartwright functions. If a > 0 and Λ ⊂ C, then by the very definition of the classical Fourier transform,

f(t) ↦ f̂(z) = ∫ e^{izt}f(t)dt,  (0.1)

The first author is supported by N.S.F. Grant No. 0201893.

The second author is supported by N.S.F. Grant No. 020699.

the family of exponential functions E_{Λ} = {e^{iλt} : λ ∈ Λ} is not complete in L^{2}(−a, a) if and only if there is a non-trivial function F from the Paley-Wiener space

PW_{a} = {f̂: f ∈ L^{2}(−a, a)},

such that F = 0 on Λ. According to Paley-Wiener’s theorem, see [33], PW_{a} can be characterized as the space of entire functions which have exponential type at most a and are square summable on the real line ℝ.

The Cartwright class Cart_{a}, a ≥ 0, consists of entire functions F of exponential type at most a satisfying a weaker integrability condition on ℝ:

log^{+} |F | ∈ L^{1}(ℝ, Π), dΠ(t) := \frac{dt}{1 + t^{2}} .

Cartwright functions are considered in detail in the monographs [29], [6], [28], [12], [25], [19]. The following Krein’s theorem [26] is important for the purpose of this discussion: an entire function is Cartwright if and only if it belongs to the Nevanlinna class in the lower and in the upper halfplanes. In a sense, Paley-Wiener spaces are to Cartwright spaces as the Hardy space H^{2} is to the Smirnov-Nevanlinna class N^{+}. The precise meaning of this analogy, which is only true up to an arbitrarily small gap in the exponential type, is a part of the deep Beurling-Malliavin theory [4]–[5].

The growth limitations (exponential type and integrability conditions on ℝ) on Paley-Wiener and Cartwright functions carry with them severe limitations on the distribution of zeros. This is a central theme of the classical theory of entire functions, and numerous results have been obtained in the study of the relation between the growth and zeros. Let us mention some principle facts. For a set Λ we denote by N(R) the number of points λ ∈ Λ satisfying |λ| < R, and by N_{±}(R) the number of points λ, |λ| < R, such that ℜλ > 0 and ℜλ < 0 respectively.

Density and symmetry of zeros. If F is a non-trivial Cartwright function and Λ is the set of all zeros, then the limits

lim_{R→∞} \frac{N_{±}(R)}{R}

exist, are finite and equal. Moreover, there exists a finite limit

v.p. ∑ \frac{1}{λ} ≡ lim_{R→∞} ∑_{|λ|<R} \frac{1}{λ}.

See, e.g., [28], [25]. Levinson’s completeness theorem. If F is a non-trivial function in PW_{a} and F = 0 on Λ (so E_{Λ} is not complete in L^{2}[−a, a]), then

lim_{R→∞} [∫_{1}^{R} \frac{N(t)}{t}dt − \frac{2a}{π}R + \frac{1}{2} log R] = −∞.

Some other sufficient conditions for completeness are known, as well as some necessary conditions, see [38], [41]. On the other hand, finding an effective general metric criterion does not seem to be realistically possible.

Beurling-Malliavin theory. The most interesting and deep result in the completeness problem is the metric characterization of the ”radius of completeness”,

R(Λ) = sup{a : E_{Λ} complete in L^{2}(−a, a)},

obtained in [4]-[5]. Beurling and Malliavin described R(Λ) as a certain ”density” d_{BM}(Λ), which is defined in a non-trivial but completely computable way. We recall the definition of d_{BM}(Λ) and discuss the Beurling-Malliavin theory in the last part of the paper.

Let us now explain how the completeness problem can be restated in terms of Toeplitz kernels.

0.2. Model spaces and Toeplitz operators. By the Paley-Wiener theorem, the Fourier transform (0.1) identifies L^{2}(0, ∞) with the Hardy space H^{2} in the upper halfplane C_{+}, and therefore it identifies L^{2}(±a, ∞) with S^{±a}H^{2}. Here and throughout the paper, S denotes the singular inner function

S(z) = e^{iz}.

It follows that the Paley-Wiener spaces have the following representation:

PW_{a} = S^{−a} [H^{2} ⊖ S^{2a}H^{2}] .

The subspace H^{2} ⊖ S^{2a}H^{2} is the so called model space of the inner function S^{2a}. More generally, one defines model spaces

K_{Θ} ≡ K[Θ] = H^{2} ⊖ ΘH^{2}

for all inner functions Θ in C_{+}; these spaces play an important role in the modern function theory and also in the spectral theory, see [34], [27]. Strictly speaking, the elements of K_{Θ} are functions in C_{+} but if Θ is a meromorphic inner function, then every element has a meromorphic continuation to the whole complex plane. In particular, the completeness problem for exponentials is exactly the problem of describing the uniqueness sets of the model spaces K[S^{2a}].

If U ∈ L^{∞}(ℝ), then the Toeplitz operator with symbol U is the map

T_{U} : H^{2} → H^{2}, F ↦ P_{+}(UF),

where P_{+} is the orthogonal projection in L^{2}(ℝ) onto H^{2}. A one line argument shows that Λ ⊂ C_{+} is a uniqueness set of K_{Θ} if and only if the kernel of the Toeplitz operator T_{U} is non-trivial, where

U = Θ̄B_{Λ}, B_{Λ} Blaschke product.

(There is a similar statement for general sets Λ ⊂ C, see Section 3.1.) The injectivity problem – to characterize symbols U such that ker T_{U} = 0 – is of interest in its own right as part of the spectral theory of Toeplitz operators, see [8], [35]. Compared with some other aspects of the theory such as invertibility problem, the injectivity problem has attracted relatively little attention. Let us mention the important paper [23] where the idea to use (invertibility) properties of Toeplitz operators in the study of bases of exponentials was introduced, see also [3].

We can now explain our goal more clearly. We would like to see if the classical results mentioned above could be extended to Toeplitz operators with more general symbols. More precisely, we’ll be considering real-analytic symbols U = e^{iγ}, γ ∈ C^{ω}(ℝ), so that infinity is the only ”singularity” of the symbol. A special case (which is in fact just as general, see [9]) is the case of symbols of the form

U = Θ̄J,

where Θ and J are meromorphic inner functions. We will show now that the injectivity problem for operators with such symbols appears as naturally as in the special case Θ = S^{2a}.

0.3. Spectral theory. Consider the Schrödinger equation

− ü + qu = λu  (0.2)

on some interval (a, b) and assume that the potential q(t) is locally integrable and a is a regular point, i.e. a if finite and q is L^{1} at a. Let us fix some selfadjoint boundary condition at b and consider the Weyl m-function

m(λ) = \frac{u̇_{λ}(a)}{u_{λ}(a)}, λ ∉ ℝ,

where u_{λ}(t) is any non-trivial solution of (0.2) satisfying the boundary condition. We will deal only with the compact resolvent case, which is equivalent to saying that m extends to a meromorphic function. Then we can define the meromorphic inner function

Θ = \frac{m − i}{m + i},

which we call the Weyl inner function associated with the potential and the fixed boundary condition at b. The transformation

f(t) ↦ F (λ) = ∫_{a}^{b} f(t)\frac{u_{λ}(t)}{u̇_{λ}(a) + iu_{λ}(a)} dt  (0.3)

identifies L^{2}(a, b) with the model space K_{Θ} in the same way as the classical Fourier transform (times S^{a}) identifies L^{2}(−a, a) with K[S^{2a}]. This allows us to interprete the completeness problem for families of solutions {u_{λ} : λ ∈ Λ} as a problem of uniqueness sets in the model space of Θ. Completeness problems of this type, particularly problems involving families of special functions, are well-known in the literature, see e.g. [20]. As we explained, they are equivalent to the invertibility problem for symbols Θ̄J, where Θ is a Weyl inner function.

Similar invertibility problems appear in connection with the uniqueness part of the inverse spectral problem. We discuss such applications in Section 3. Here we only mention that the results represent a rather broad generalization of such wellknown facts as Borg’s two spectra theorem [7]: two different spectra of a Schrodinger operator with compact resolvent determine the operator uniquely, and Hochstadt-Liberman theorem [21] that states that a regular Schrödinger operator on a finite interval is determined by its spectrum and the potential on one half of the interval. In the language of inner functions, the corresponding problems can be stated as follows.

Given a meromorphic inner function Θ and one of its factors Ψ, the problem is to decide whether this factor and the set {Θ = 1} determine Θ uniquely. The second problem is to describe defining sets of a given meromorphic inner function Φ. We say that Λ ⊂ ℝ is defining if

Φ̃ = Φ, arg Φ̃ = arg Φ on Λ  ⇒  Φ̃ ≡ Φ.

0.4. Content of the paper.

Section 1. Meromorphic inner functions and spectral theory.

1.1–1.4: We recall standard facts concerning meromorphic inner functions, their model spaces, and Weyl-Titchmarsh functions of second order selfadjoint differential operators.

1.5: We discuss the modified Fourier transform (0.3). In the case of regular operators, this is essentially the usual Weyl-Titchmarsh transform, but the construction is probably new in the (more interesting) singular case.

1.6–1.8: Basics of de Branges functions and associated spaces of Paley-Wiener and Cartwright type.

Section 2. Toeplitz kernels.

2.1-2.4: We define the kernels and state some general (mostly known) facts. As an example, we give a Toeplitz kernel interpretation of a standard asymptotic formula for solutions of a regular Schrödinger equation.

2.5–2.6: Basic criterion for non-triviality of a Toeplitz kernel with real analytic symbol. For instance, the kernel is non-trivial in the Smirnov-Nevanlinna class if and only if the argument γ of the symbol has a representation γ = −α + h̃, where α ∈ C^{ω} is an increasing function and h ∈ L_{Π}^{1}. Though this observation is very simple (and its versions in the non-analytic case are well known), the criterion turns out to be quite workable, and the rest of the paper is mostly the study and applications of this criterion.

2.7–2.8: Kolmogorov’s type criterion. This is a special case where the symbol is H̄/H, H is an outer function, and the kernel is a priory finite dimensional. This situation is typical when we explicitly know the de Branges functions. Another useful example is the twin inner function theorem: if {Θ = 1} = {J = 1}, then ker T_{Θ̄J} = 0

2.9–2.11: General form of Levinson’s completeness theorem. We obtain a sufficient condition for triviality of a Toeplitz kernel that improves Levinson’s theorem (and other similar results) even in the classical situation. The key ingredient of the proof is the Titchmarsh-Uly’anov theorem involving the so called A-integrals.

Section 3. Some applications.

3.1–3.2: Completeness and minimality problem, and uniqueness sets of the model and de Branges spaces.

3.3–3.4: Distribution of zeros of functions in Cartwright-de Branges spaces. In particular, we give a new proof of the density and symmetry result mentioned in Section 0.1, which is based on our basic criterion and the Titchmarsh-Uly’anov theorem.

3.5–3.7: Applications to the mixed data spectral problem stated in Section 0.3. For the inner function version of the Hochstadt-Liberman problem we establish some necessary and some sufficient conditions in terms of Toeplitz kernels. We also give a spectral theory interpretation of these conditions indicating stronger versions of practically all known results in this area.

3.8–3.9: Remarks on defining sets of inner functions and regular Schrödinger operators.

Section 4. Beurling-Malliavin theory.

4.1-4.2: Multiplier theorems. First we state a multiplier theorem for Toeplitz kernels in H^{p}-spaces. Then we discuss some consequences of the Beurling-Malliavin multiplier theorem for Toeplitz kernels in the Smirnov-Nevanlinna class. We make no comments on the proof of the Beurling-Malliavin multiplier theorem itself. The presence of the Dirichlet space condition remains the most amazing feature of the theory.

4.3–4.6: Second Beurling-Malliavin and little multiplier theorems. For symbols U = e^{iγ} with γ′ > −const, we present a complete proof of the metric criterion for (non-)triviality of a Toeplitz kernel in the Smirnov-Nevanlinna class up to a gap S^{±ϵ}. The proof is of course not totally original but our version, we believe, is better fit for generalizations.

4.7–4.9: We discuss possible generalizations of the Beurling-Malliavin theory. We mention partial results, examples, and indicate applications in the case γ′(t) > −const |t|^{a}.

1. Meromorphic inner functions and spectral theory

Function Theory in the halfplane

1.1. Basic notations. C_{+} is the upper half plane {ℜz > 0}. For general references concerning Hardy-Nevanlinna theory in C_{+} see [16] and [35].

We use the standard notation H^{p} = H^{p}(C_{+}), 0 < p ≤ ∞, for the Hardy spaces, and N^{+} = N^{+}(C_{+}) for the V. I. Smirnov (or Smirnov-Nevanlinna) class in C_{+}. The elements in N^{+} are ratios G/H, where G, H ∈ H^{∞} and H is an outer function. Functions in N^{+} have angular boundary values (almost everywhere) on the real line. As a general rule, we identify functions in the halfplane with their boundary values on ℝ. In this sense, we have

H^{p} = N^{+} ∩ L^{p}(ℝ),

and

F ∈ N^{+} ⇒ log |F | ∈ L_{Π}^{1} ≡ L^{1}(ℝ, Π),

where Π is the Poisson measure

dΠ(t) = \frac{dt}{1 + t^{2}} .

If h ∈ L_{Π}^{1} is a real-valued function, then its Schwarz integral is

Sh(z) = \frac{1}{πi} ∫ [\frac{1}{t − z} − \frac{t}{1 + t^{2}} ] h(t)dt.

The real and the imaginary parts of Sh are the Poisson and the conjugate Poisson integrals of h:

Sh = Ph + iQh.

Outer functions are functions of the form

H = e^{Sh}, h ∈ L_{Π}^{1};

note that H ∈ N^{+} and H has modulus e^{h} on ℝ. Every function F in N^{+} has a unique factorization F = IH, where H is the outer function with modulus |F | on ℝ and I is an inner function, i.e. I ∈ H^{∞} and |I| = 1 on ℝ.

The Hilbert transform of h ∈ L_{Π}^{1} is the angular limit of Qh, so the outer function e^{Sh} is equal to e^{h+ih̃} on ℝ. The Hilbert transform can also be defined as a singular integral:

h̃(x) = \frac{1}{π} v.p. ∫ [\frac{1}{x − t} + \frac{t}{1 + t^{2}} ] h(t)dt.

For further references, we recall some properties of the Hilbert transform. If both h and g = h̃ are in L_{Π}^{1}, then then g̃ = −h + const, i.e.

Sh̃ = −iSh + iSh(i).

If h ∈ L_{Π}^{1}, then h̃ ∈ L_{Π}^{o(1,∞)} (the weak L^{1} space), i.e.

Π\{|h̃| > A\} = o (\frac{1}{A}) , A → ∞,

in particular h̃ ∈ L_{Π}^{p} for all p < 1.

1.2. Meromorphic inner functions and Herglotz functions. A meromorphic inner function is an inner function Θ in C_{+} which has a meromorphic extension to C. Such a function can be characterized by parameters (a, Λ) in the canonical (Riesz-Smirnov) factorization

Θ = B_{Λ}S^{a},  (1.1)

where a ≥ 0, and Λ is a discrete set (possibly with multiple points) in C_{+} satisfying the Blaschke condition

∑ \frac{ℑλ}{1 + |λ|^{2}} < ∞;

B_{Λ} denotes the corresponding Blaschke product and S^{a}(z) = e^{iaz}. Let us mention an obvious but important property of meromorphic inner functions:

Θ = e^{iθ} on ℝ, θ is a real analytic, increasing function.

A meromorphic Herglotz function is a meromorphic functions m such that

ℑm > 0 in C_{+}, m(z̄) = \overline{m(z)}.

One can establish a 1-to-1 correspondence between meromorphic inner and Herglotz functions by means of the equations

m = i\frac{1 + Θ}{1 − Θ},  Θ = \frac{m − i}{m + i}.  (1.2)

Meromorphic Herglotz functions (and therefore inner functions) can be described by parameters (b, c, μ) in the Herglotz representation

m(z) = bz + c + iSμ,  (1.3)

where b ≥ 0, c ∈ ℝ, and μ is a positive discrete measure on ℝ satisfying

∫ \frac{dμ(t)}{1 + t^{2}} < ∞.

It is convenient to interprete the number πb as a point mass of μ at infinity. In the case m = m_{Θ}, see (1.2), we call this extended measure μ_{Θ} the spectral (or Herglotz) measure of Θ. By definition, the (point) spectrum of Θ is the set

σ(Θ) = supp μ_{Θ} = {Θ = 1} or {Θ = 1} ∪ {∞},

and by residue calculus we have

μ_{Θ}(t) = \frac{2π}{|Θ′(t)|},  t ∈ σ(Θ).  (1.4)

The following equivalent conditions are necessary and sufficient for μ_{Θ}(∞) ≠ 0, see e.g. [36]:

(i) Θ − 1 ∈ H^{2};  (ii) Θ(∞) = 1, ∃Θ′(∞);  (iii) ∑ ℑλ < ∞.

In (ii), Θ(∞) and Θ′(∞) mean the angular limit and angular derivative at infinity:

Θ(∞) = lim_{y→+∞} Θ(iy), Θ′(∞) = lim_{y→+∞} y^{2}Θ′(iy),

and in (iii) we also require that the singular factor is trivial.

Riesz-Smirnov and Herglotz parametrizations (1.1)-(1.3) reflect two different structures – multiplicative and convex – in the set of inner functions. These structures are related in a non-trivial and intriguing way. For example, the middle point of the segment [Θ_{1}, Θ_{2}], i.e. the inner function such that its Herglotz measure is the average of μ_{Θ_{1}} and μ_{Θ_{2}}, is the function

Θ = \frac{Θ_{1} + Θ_{2} − 2Θ_{1}Θ_{2}}{2 − Θ_{1} − Θ_{2}} ,

and we observe that

Ψ|Θ_{1}, Ψ|Θ_{2}  ⇒  Ψ|Θ,

where the notation Ψ|Θ for two inner functions means that Ψ is a factor of Θ, i.e. Θ/Ψ is also an inner function.

1.3. Model spaces. The H^{2}-model space of an inner function Θ,

K_{Θ} ≡ K[Θ] = H^{2} ⊖ ΘH^{2} = H^{2} ∩ ΘH̄^{2},

is a Hilbert space with reproducing kernel:

k_{λ}^{Θ}(z) = \frac{1}{2πi} \frac{1 − \overline{Θ(λ)}Θ(z)}{λ̄ − z},  λ ∈ C_{+}.  (1.5)

If Θ is meromorphic, then all elements of K_{Θ} are meromorphic, and one can extend (1.5) to all λ ∈ ℝ.

The monograph [34] provides a comprehensive study of model spaces. One of the important facts of the theory is the fpllowing Plancherel theorem (see Clark’s paper [10] for the case of general inner functions): the restriction map

C_{Θ} : f ↦ f|_{σ(Θ)}  (1.6)

is a unitary operator K_{Θ} → L^{2}(μ_{Θ}).

We also define the model spaces in the Smirnov class and in general Hardy spaces:

K_{Θ}^{+} = {F ∈ N^{+} ∩ C^{ω}(ℝ) : ΘF̄ ∈ N^{+}},

and

K_{Θ}^{p} = K_{Θ}^{+} ∩ L^{p}(ℝ).

If p ≥ 1, we can drop the requirement F ∈ C^{ω}(ℝ) by Morera’s theorem.

Second order differential operators

1.4. Weyl inner functions. Meromorphic inner functions appear in the theory of 2nd order selfadjoint differential operators with compact resolvent. We will only discuss the case of Schrödinger operators though similar theories exist for general canonical systems. See [32] and [30] for the basics of the spectral theory.

Let q be a real locally integrable function on (a, b). We always assume that selfadjoint operators associated with the differential operation u ↦ −ü+qu have compact resolvent. We suppose that a is a regular point but we allow b to be infinite and/or singular. Let us fix a selfadjoint boundary condition β at b; for example, β means u ∈ L^{2} at b in the limit point case. The Weyl-Titchmarsh m-function of (q; b, β) evaluated at a,

m(λ) = m_{b,β}^{a}(λ), λ ∈ C,

is defined by the formula

m(λ) = \frac{u̇_{λ}(a)}{u_{λ}(a)},

where u_{λ}(·) is a non-trivial solution of the Schrödinger equation satisfying the boundary condition at b. It is well-known that m is a Herglotz function, and therefore we can define the corresponding inner function Θ_{b,β}^{a} by (1.2). We will call Θ_{b,β}^{a} the Weyl (or Weyl-Titchmarsh) inner function of q.

Similarly, if b ∈ ℝ is a regular point and α is a selfadjoint boundary condition at a ∈ [−∞, b), we can consider the m-function of (q; a, α) evaluated at b,

m_{a,α}^{b}(λ) = −\frac{u̇_{λ}(b)}{u_{λ}(b)}

(mind the sign!) and define the corresponding Weyl inner function Θ_{a,α}^{b}.

Example. The Weyl inner functions of the potential q ≡ 0 on [0, 1] with Dirichlet and, respectively, Neumann boundary conditions at a = 0 are

Θ_{D}(λ) = \frac{\sqrt{λ} cos \sqrt{λ} + i sin \sqrt{λ}}{\sqrt{λ} cos \sqrt{λ} − i sin \sqrt{λ}},  Θ_{N}(λ) = \frac{\sqrt{λ} sin \sqrt{λ} − i cos \sqrt{λ}}{\sqrt{λ} sin \sqrt{λ} + i cos \sqrt{λ}}. (1.7)

(The m-functions are m_{D}(λ) = −\sqrt{λ} cot \sqrt{λ}, and and m_{N}(λ) = \sqrt{λ} tan \sqrt{λ}.) More generally, for ν ≥ −1/2 consider the potential

q(t) = \frac{ν^{2} − \frac{1}{4}}{t^{2}} on (0, 1),

and let the boundary condition α at a = 0 be satisfied by the solution

u_{λ}(t) = \sqrt{t}J_{ν}(t\sqrt{λ})

of the Schrödinger equation. For example, if ν = −1/2 then α = (N), and if ν = 1/2 then α = (D), and we have the limit point case if ν ≥ 1. J_{ν} is of course the standard notation for the Bessel function of order ν. Since

u_{λ}(1) = J_{ν}(\sqrt{λ}),  u̇_{λ}(1) = \frac{1}{2}J_{ν}(\sqrt{λ}) + \sqrt{λ}J′_{ν}(\sqrt{λ}),

the corresponding Weyl inner function is

Θ_{ν}(λ) = \frac{\sqrt{λ}J′_{ν}(\sqrt{λ}) + (1/2 + i)J_{ν}(\sqrt{λ})}{\sqrt{λ}J′_{ν}(\sqrt{λ}) + (1/2 − i)J_{ν}(\sqrt{λ})}.  (1.8)

In particular, we have Θ_{−1/2} = Θ_{N} and Θ_{1/2} = Θ_{D}.

One can give many other similar examples involving special functions. We will continue to discuss Bessel inner function in Sections 1.6 and 3.7. Our goal is to illustrate certain constructions in the singular case as opposed to the regular case, which is well presented in the literature.

1.5. Modified Fourier transform. Let Θ = Θ_{b,β}^{a} be the Weyl-Titchmarsh inner function of a potential q defined in the previous section. We will construct a unitary operator L^{2}(a, b) → K_{Θ}, which is a modification of the Weyl-Titchmarsh Fourier transform. We modify the usual construction so that the case of a singular endpoint b could be included.

For every z ∈ C we choose a non-trivial solution u_{z}(t) of the Schrödinger equation satisfying the boundary condition β. (For real z such a solution exists because of the compact resolvent assumption). If z ∈ C_{+} ∪ ℝ, then the solution

w_{z}(t) = \frac{u_{z}(t)}{u̇_{z}(a) + iu_{z}(a)}

does not depend on the choice of u_{z}, and w_{z} ∈ L^{2}(a, b). The transform W is defined as follows:

W : f(t) ↦ F (z) = ∫_{a}^{b} f(t)w_{z}(t)dt, (z ∈ C_{+} ∪ ℝ).  (1.9)

To state the main result we introduce the dual reproducing kernel of the model space K_{Θ}. For λ ∈ C_{+} ∪ ℝ we define

k_{λ}^{∗}(z) = \frac{1}{2πi} \frac{Θ(z) − Θ(λ)}{z − λ},  (z ∈ C_{+} ∪ ℝ),  (1.10)

so we have

Θ̄k_{λ}^{Θ} = \overline{k_{λ}^{∗}} on ℝ,

and k_{λ}^{∗} ∈ K_{Θ}. Note that if λ ∈ ℝ, then k_{λ}^{∗} = const k_{λ}^{Θ}.

Theorem. The modified Fourier transform W is (up to a factor \sqrt{π}) a unitary operator L^{2}(a, b) → K_{Θ}. Furthermore, we have

Ww_{λ} = πk_{λ}^{∗}, Ww̄_{λ} = πk_{λ}  (λ ∈ C_{+} ∪ ℝ).  (1.11)

Proof: The formulae (1.11) follow from the Lagrange identity

(z − λ) ∫_{a}^{b} u_{λ}u_{z} = u_{λ}(a)u̇_{z}(a) − u̇_{λ}(a)u_{z}(a).

(The Wronskian at b is zero because the two solutions satisfy the same boundary conditions.) The rest is straightforward:

(w̄_{λ}, w̄_{μ})_{L^{2}} = ∫_{a}^{b} w_{μ}w̄_{λ} = Ww̄_{λ}(μ) = πk_{λ}(μ) = π(k_{λ}, k_{μ})_{K_{Θ}},

etc. □

Note that Weyl inner functions of Schrödinger operators have no point masses at infinity, so if Θ = Θ_{b,β}^{a}, then

σ(Θ) = σ(q, D, β), σ(−Θ) = σ(q, N, β).

Here σ(q, D, β) means the spectrum of the Schrödinger operator with potential q, Dirichlet boundary condition at a, and boundary condition β at b. More generally, for α ∈ ℝ let α denote the following selfadjoint boundary condition at a regular endpoint a:

cos \frac{α}{2}u(a) + sin \frac{α}{2}u̇(a) = 0.  (1.12)

Then

σ(e^{−iα}Θ) = σ(q, α, β).

By definition, the spectral measure of the Schrödinger operator (q, α, β) is the Herglotz measure of the inner function e^{−iα}Θ.

Corollary. Let Θ = Θ_{b,β}^{a}. The composition of the modified Fourier transform and the Plancherel-Clark operator (1.6),

L^{2}(a, b) ⟶^{W} K_{Θ} ⟶^{C_{Θ}} L^{2}(μ_{Θ}),

is a unitary operator; it provides a spectral representation of the Schrödinger operator (q, D, β).

Entire functions

1.6. de Branges functions. Following [28] we say that an entire function E is of Hermite-Biehler class (HB) if E has no real zeros and

z ∈ C_{+} ⇒ |E(z̄)| < |E(z)|.

Every E ∈(HB) defines a meromorphic inner function

Θ_{E} = \frac{E^{#}}{E} , E^{#}(z) := \overline{E(z̄)}.

Conversely, given an inner function Θ, any E ∈ (HB) satisfying Θ = Θ_{E} is called a de Branges function of Θ.

It can be shown, see [12], that every meromorphic inner function has at least one de Branges function. In some cases one can construct de Branges functions explicitly.

Examples. (i) Let (a, b) be a finite interval and q ∈ L^{1}(a, b). Given a selfadjoint boundary condition (1.12) at a, let u_{λ}(t) denote the solution of the initial value problem

u_{λ}(a) = − sin \frac{α}{2},  u̇_{λ}(a) = cos \frac{α}{2}.  (1.13)

for the Schrödinger equation. Then

E(λ) = −u̇_{λ}(b) + iu_{λ}(b)  (1.14)

is a de Branges function of the Weyl inner function Θ = Θ_{a,α}^{b}. Indeed, we have

Θ(λ) = \frac{−u̇_{λ}(b) − iu_{λ}(b)}{−u̇_{λ}(b) + iu_{λ}(b)},

and the functions λ ↦ u_{λ}(b) and λ ↦ u̇_{λ}(b) are entire because of the fixed initial conditions; clearly, they can not be both zero at the same point λ ∈ ℝ.

(ii) Consider now the Bessel inner functions Θ_{ν}, see (1.8). Note that the above construction does not apply in the singular case. From the theory of Bessel’s functions we know that

J_{ν}(z) = z^{ν}G_{ν}(z),

where G_{ν} is an even real entire functions and G_{ν}(0) ≠ 0. We also introduce an even real entire function

F_{ν}(z) = zG′_{ν}(z).

Since zJ′_{ν} = z^{ν}(νG_{ν} + F_{ν}), we have

Θ_{ν}(λ) = \frac{F_{ν}(\sqrt{λ}) + (1/2 + ν + i)G_{ν}(\sqrt{λ})}{F_{ν}(\sqrt{λ}) + (1/2 + ν − i)G_{ν}(\sqrt{λ})}.

The function

E_{ν}(λ) := F_{ν}(\sqrt{λ}) + (1/2 + ν − i)G_{ν}(\sqrt{λ})

does not vanish at λ = 0 and therefore it has no zeros on ℝ. It follows that E_{ν} is a de Branges function of Θ_{ν}. Similar considerations work for other special functions.

(iii) Some more elementary (but important) examples include the following. If a > 0, then E(z) = e^{−iaz} is a de Branges function of Θ = S^{2a}. Polynomials with all roots in C_{−} are de Branges functions of finite Blaschke products.

1.7. Spaces of entire functions. We first define the Cartwright-de Branges space B^{+}(E) of entire functions associated with a Hermite-Biehler function E:

B^{+}(E) = {F : F/E, F^{#}/E ∈ N^{+}(C_{+})} .

Proposition. B^{+}(E) = EK^{+} [Θ_{E}] .

Proof: If G ∈ K_{Θ}^{+}, then its meromorphic extention to C_{−} is equal to H^{#}/Θ^{#} for some H ∈ N^{+}(C_{+}). Since E = Θ^{#}E^{#} in C_{−}, the function

F = \begin{cases}E(z) G(z), z ∈ C_{+} \\ E^{#}(z)H^{#}(z), z ∈ C_{−} \end{cases}

is entire, and F ∈ B^{+}(E). The opposite direction is similar. □

The special case E = S^{−a} gives Cartwright spaces

Cart_{a} = B^{+} (S^{−a}) = S^{−a}K^{+} [S^{2a}] , (a ≥ 0).

Next we define the de Branges space associated with E ∈ (HB), see [12]:

B(E) = B^{+}(E) ∩ L^{2} (|E(x)|^{−2}dx) = EK [Θ_{E}] .

The special case E = S^{−a} gives the Paley-Wiener spaces

PW_{a} = B (S^{−a}) = Cart_{a} ∩ L^{2}(ℝ), (a > 0).

De Branges space B(E) has a natural Hilbert space structure so that the multiplication operator

E : K [Θ_{E}] → B(E)

is an isometry. We denote by K_{λ}^{E}, (λ ∈ C), the reproducing kernel of B(E). Theorem 1.5 has the following corollary, which is a counterpart of the Paley-Wiener theorem concerning the classical Fourier transform. Recall that W denotes the modified Fourier transform (1.9).

Corollary. Let E be a de Branges function of the Weyl inner function Θ_{b,β}^{a} associated with a potential q on (a, b). Then the map

F : L^{2}(a, b) → B(E), f ↦ E · Wf,

is a unitary operator. Furthermore, we have

Fu_{λ} = const KE ; λ̄, λ ∈ C,

where u_{λ} is any non-trivial solution of the Schrödinger equation satisfying the boundary condition β.

In the regular case a, b ∈ ℝ and q ∈ L^{1}(a, b), the map F is precisely the Weyl-Titchmarsh transform

f ↦ ∫_{a}^{b} f(t)u_{λ}(t)dt,

where the solutions u_{λ}(t) are normalized by initial conditions (1.13) and the de Branges function is given by (1.14).

The classical Fourier transform (0.1) originates from the 1-st order selfadjoint operator u ↦ −iu′. Alternatively, it can be related to the Weyl-Titchmarsh transforms corresponding to q ≡ 0 by a general construction which we describe below, cf. [14].

1.8. Square root transformation. It is well-known that if m is a Herglotz function such that

0 < m < +∞ on ℝ_{−},

then m^{∗}(λ) = λm(λ^{2}) is again a Herglotz function. If Θ = (m − i)/(m + i), then the inner function corresponding to m^{∗} is

Θ^{∗}(z) = \frac{(z + 1)Θ(z^{2}) + (z − 1)}{(z − 1)Θ(z^{2}) + (z + 1)}

We call Θ^{∗} the square root transform of Θ. Suppose now that E = A + iB is a de Branges function of Θ, where A and B are real entire functions, and also suppose B(0) ≠ 0. Then

E^{∗}(z) = zA(z^{2}) + iB(z^{2})

is a de Branges function of Θ^{∗}.

Example. Let q ∈ L^{1}[0, 1] be such that the operator L(q, D, N) ≥ 0. Consider the function

m(λ) = \frac{u(λ)}{u̇(λ)} := \frac{u_{λ}(1)}{u̇_{λ}(1)},

where u_{λ} is the solution of the Schrödinger equation with initial conditions u_{λ}(0) = 0 and u̇_{λ}(0) = 1. (In other words, m is the Herglotz function of −Θ_{0,D}^{1}.) Then we have E = u + iu̇ and therefore

E^{∗}(z) = zu(z^{2}) + iu̇(z^{2}).

In particular, if q ≡ 0, then E^{∗}(z) = ie^{−iz}, and we get the classical Paley-Wiener space.

2. Toeplitz kernels

Some generalities

2.1. Definition of Toeplitz kernels. Recall that to every U ∈ L^{∞}(ℝ), there corresponds the Toeplitz operator T_{U} : H^{2} → H^{2}. We need to consider only the case of unimodular symbols

U = e^{iγ}, γ : ℝ → ℝ,

and we will concentrate on the question whether the Toeplitz kernel

N[U] = ker T_{U}

is trivial or non-trivial. The best known situation is when γ ∈ C(ℝ) and there exist finite limits γ(±∞). (This corresponds to the case of piecewise continuous symbols in the theory on the unit circle.) If we denote

δ = γ(+∞) − γ(−∞),

then

N[U] = 0 if δ > −π, N[U] ≠ 0 if δ < −π.

(If δ = −π, then either case is possible.)

Along with H^{2}-kernels, we define Toeplitz kernels in the Smirnov class,

N^{+}[U] = {F ∈ N^{+} ∩ L_{loc}^{1}(ℝ) : ŪF̄ ∈ N^{+}},

and in all Hardy spaces,

N^{p}[U] = N^{+}[U] ∩ L^{p}(ℝ), (0 < p ≤ ∞).

These definitions are oriented to studying the case where ∞ is the only ”singularity” of the symbol. In particular, if Θ is a meromorphic inner function, then N^{+}[Θ̄] = K_{Θ}^{+} and N^{p}[Θ̄] = K_{Θ}^{p}.

We use the notation b for the Blaschke factor

b(z) = \frac{i − z}{i + z} .

The argument 2 arctan(x) of b increases from −π at −∞ to +π at +∞. One can characterize the dimension of a Toeplitz kernel by multiplying the symbol by integer powers of b.

Lemma. For n ∈ N, dim N^{p}[U] = n + 1 iff dim N^{p}[b^{n}U] = 1.

Proof: If, for instance, dim N^{p}[U] ≥ 2, then we can find an F ∈ N^{p}[U] such that F (i) = 0, and so b̄F ∈ N^{p}[bU] and dim N^{p}[bU] ≥ 1. In the opposite direction, if G ∈ N^{p}[bU], then both G and bG are in N^{p}[U]. □

One can also consider fractional powers of b:

b^{s}(x) = exp{2si arctan x}, (s ∈ ℝ).

The identity

b̄^{s}(1 − b)^{s} = (b̄ − 1)^{s}

shows that N^{∞}[b̄^{s}] ≠ 0 for s ≥ 0. It follows that for every U and p > 0 there is a critical value s_{∗} ∈ ℝ ∪ {±∞} such that

N^{p}[b̄^{s}U] ≠ 0 if s > s_{∗}, N^{p}[b̄^{s}U] = 0 if s < s_{∗}.

One can interpret s_{∗} as a fractional and possibly negative ”dimension” of the kernel. For example, ”dim”N[1] = −1/2 , but ”dim”N[Θ̄_{N}Θ_{D}] = −1/4 for the Dirichlet and Neumann inner functions (1.7), see Section 2.7 below.

2.2. Basic criterion. The following well-known observation is extremely simple but quite useful. In fact, most of our further constructions are built upon this lemma.

Lemma. N^{p}[U] ≠ 0 iff the symbol has the following representation:

U = Φ̄\frac{H̄}{H} ,

where H ∈ H^{p} ∩ L_{loc}^{1}(ℝ) is an outer function and Φ is an inner function.

Proof: If UF = Ḡ, then |F | = |G| on ℝ. Consider the inner-outer factorization: F = F_{i}F_{e} and G = G_{i}G_{e}. We have F_{e} = G_{e}, and

U = (F̄_{i}Ḡ_{i})\frac{F̄_{e}}{F_{e}} .

The converse is obvious. □

Corollary. If γ ∈ L^{∞} ∩ C(ℝ), then ∃p > 0, ”dim”N^{p}[e^{iγ}] > −∞.

A more precise statement is

∥γ∥_{∞} < \frac{π}{p}  ⇒  N^{p}[b̄^{2/p}e^{iγ}] ≠ 0,

which follows from the Smirnov-Kolmogorov estimate

∥h∥_{∞} < \frac{π}{2}  ⇒  eh̃ ∈ L_{Π}^{1},

and from the construction of outer functions.

Of course, instead of ∥γ∥_{∞} < ∞ we can only require that γ be the sum of a decreasing and a BMO functions, and we don’t need continuity if p ≥ 1. It is also important to realize that p can not be arbitrary in the statement of the corollary. For instance, it is easy to construct γ ∈ C^{ω}(ℝ) such that ∥γ∥_{∞} = π/2 but N[b̄^{n}e^{iγ}] = 0 for all n > 0.

2.3. Sufficient conditions for dim N^{p}[u] < ∞. The following statement is a version of Coburn’s lemma, which states that either ker T_{U} = 0 or ker TŪ = 0.

Lemma. If 1/p + 1/q > s, then

N^{q}[Ū] ∩ L_{loc}^{2}(ℝ) ≠ 0 ⇒ N^{p}[b̄^{s}U] ∩ L_{loc}^{2}(ℝ) = 0.

Proof: Suppose both kernels are non-trivial:

ŪF_{1} = Ḡ_{1},  b̄^{s}UF_{2} = Ḡ_{2}

for some F_{1}, G_{1} ∈ H^{q} and F_{2}, G_{2} ∈ H^{p}. Then

(i + z)^{s}F_{1}(z)F_{2}(z) = (i − z̄)^{s}Ḡ_{1}(z̄)Ḡ_{2}(z̄) on ℝ,

so we have an entire Cartwright function with at most a polynomial growth along iℝ. The growth at +i∞ is

y^{s}y^{−1/q}y^{−1/p},

so the entire function is zero if 1/p + 1/q > s. □

Corollary. If γ is the sum of an increasing and a bounded functions, then for all p > 0, dim N^{p}[e^{iγ}] < ∞.

Corollary. If U = H̄/H and H is an outer function such that

∃q > 0 ∃N, \frac{1}{H} ∈ L^{q} (\frac{dt}{1 + |t|^{N}} ) ,

then for all p > 0, dim N^{p}[U] < ∞.

2.4. Trivial factors. The following lemma is obvious.

Lemma. If V = H̄/H with H^{±1} ∈ H^{∞}, then N^{p}[UV] ≠ 0 iff N^{p}[U] ≠ 0.

We will call such functions V trivial factors of the symbol. Examples:

(i) If B_{1} and B_{2} are finite Blaschke products of the same degree, then N^{p}[U] = 0 iff N^{p}[B̄_{1}B_{2}U] = 0. (ii) If we modify a smooth symbol on a compact part of ℝ, then this does not affect (non-)triviality of the Toeplitz kernels. Thus the injectivity property depends only on the behavior of a smooth symbol at infinity. (iii) It is shown in [9] that up to a trivial factor every unimodular function is the ratio of two inner functions.

More relevant to the subject of the paper is our next

Example: Weyl inner functions of regular operators.

Let q ∈ L^{1}[0, 1] and let α be a non-Dirichlet selfadjoint boundary condition at a = 0. Denote by Θ the Weyl inner function of (q, α) computed at b = 1, i.e. Θ = Θ_{α,a}^{b} in the notation of Section 1.4. We want to compare Θ with the ”Neumann” inner function Θ_{N}, see (1.7), which corresponds to the special case q ≡ 0 and α = (N).

Claim. The ratio Θ/Θ_{N} is a trivial factor.

In other words, in all problems involving Toeplitz kernels we are free to replace regular potentials with the trivial potential, and any non-Dirichlet boundary condition with the Neumann condition.

Proof: We will express the ratio of the Weyl inner functions in terms of their de Branges functions. The de Branges function of Θ_{N} is

E_{N}(λ) = cos \sqrt{λ} − i\sqrt{λ} sin \sqrt{λ},

and by (1.14) the de Branges function of Θ is

E(λ) = −u̇(λ) + iu(λ), u(λ) := u_{λ}(1), u̇(λ) := u̇_{λ}(1),

where u_{λ}(t) is the solution of the Schrödinger equation with boundary condition α and initial value u_{λ}(0) = 1. We have

\frac{Θ}{Θ_{N}} = \frac{H̄}{H} , H = \frac{E}{E_{N}} .

Since both de Branges functions are outer in C_{+}, all we need to check is that |E| ≍ |E_{N}| on ℝ. To this end we can use the standard asymptotic formulae for solutions of a regular Schrödinger equation, see e.g. [30]:

|u(λ) − cos \sqrt{λ}| = O (\frac{1}{\sqrt{λ}}) ,  |u̇(λ) + \sqrt{λ} sin \sqrt{λ}| = O(1),  (λ → ±∞).

For instance if q ≡ 0 but α ≠ (N), then

u(λ) = \frac{cos(\sqrt{λ} + ψ(λ))}{cos ψ(λ)}, \sqrt{λ} tan ψ(λ) = cot \frac{α}{2},

and the asymptotic is obvious.

If λ → +∞, then

|E(λ)|^{2}, |E_{N}(λ)|^{2} ≍ [cos \sqrt{λ} + O(λ^{−1/2})]^{2} + λ[sin \sqrt{λ} + O(λ^{−1/2})]^{2} := I + II,

and we consider three cases:

if | sin \sqrt{λ}| ≲ λ^{−1/2}, then I ≍ 1 and 0 ≤ II ≲ 1, so I + II ≍ 1;

if | cos \sqrt{λ}| ≲ λ^{−1/2}, then sin^{2} \sqrt{λ} ≈ 1 and both |E|^{2} and |E_{N}|^{2} are ≍ λ;

if | sin \sqrt{λ}| and | cos \sqrt{λ}| are ≫ λ^{−1/2}, then I + II ≍ cos^{2} \sqrt{λ} + λ sin^{2} \sqrt{λ}.

The estimates for λ → −∞ are even easier. □

Toeplitz kernels with real analytic symbols

From now on we will be considering unimodular functions with real analytic arguments, U = e^{iγ}, γ ∈ C^{ω}(ℝ). In this case, all elements of the Toeplitz kernels are also real analytic on ℝ.

Lemma. If γ ∈ C^{ω}(ℝ), then N^{+}[e^{iγ}] ⊂ C^{ω}(ℝ).

Proof: Let F ∈ N^{+}[U] and let G_{−} be the analytic extension of UF to C_{−}. Since U ≠ 0 in a neighborhood of ℝ and F = U^{−1}G_{−} on ℝ, F can be extended to a neighborhood of ℝ. □

2.5. Basic criterion in N^{+}.

Proposition. Let γ ∈ C^{ω}(ℝ). Then N^{+}[e^{iγ}] ≠ 0 iff γ has a representation

γ = −α + h̃,

where α ∈ C^{ω}(ℝ) is an increasing function and h ∈ L_{Π}^{1}.

Proof: We first observe that N^{+}[u] ≠ 0 iff

U = Φ̄\frac{H̄}{H}  on ℝ  (2.1)

or some outer function H ∈ C^{ω}(ℝ) that does not vanish on ℝ, and some meromorphic inner function Φ. Indeed, suppose N^{+}[U] ≠ 0. Reasoning as in Lemma 2.2, we see that

U = Ī\frac{F̄}{F} ,

for some meromorphic inner function I and an outer function F ∈ C^{ω}(ℝ). The outer function may have zeros on the real line. Suppose the zeros are simple. Take any meromorphic inner function J such that {J = 1} = {F = 0}. Then the outer function

H = \frac{F}{1 − J}  (2.2)

is zero-free on ℝ and

U = Ī· \frac{1 − J̄}{1 − J} · \frac{H̄}{H} = −ĪJ̄\frac{H̄}{H} := Φ̄\frac{H̄}{H} .

If F has multiple zeros, then we simply repeat this reasoning taking care of the convergence.

Next we restate (2.1) in terms of the arguments of the involved functions. Since H is an outer function, it has the following representation:

H = e^{−(h+ih̃)/2}, h ∈ L_{Π}^{1},

and since H is zero free we have h̃ ∈ C^{ω}(ℝ). It follows that γ = −ϕ + h̃, where ϕ is a continuous argument of Φ. Since ϕ is strictly increasing, this gives the ”only if” part of the theorem. To prove the ”if” part, we observe that given an increasing function α, we can find an inner function with argument ϕ such that

β := α − ϕ ∈ L^{∞}(ℝ),

so

γ = −α + h̃ = −ϕ − β + h̃ = −ϕ + h̃_{1}, h_{1} := h + β̃.
□

2.6. Basic criterion in H^{p}.

Proposition. Let U = e^{iγ} with γ ∈ C^{ω}(ℝ). Then N^{p}[U] ≠ 0 iff

U = Φ̄\frac{H̄}{H} ,

where H is an outer function in H^{p} ∩C^{ω}(ℝ), H ≠ 0 on ℝ, and Φ is a meromorphic inner function. Alternatively, N^{p}[U] ≠ 0 iff

γ = −ϕ + h̃,  h ∈ L_{Π}^{1}, e^{−h} ∈ L^{p/2}(ℝ),  (2.3)

where ϕ is the argument of some meromorphic inner function.

To prove the statement we just repeat the previous proof using the following lemma, in which we construct the Herglotz measure of a meromorphic inner function J so that the function H in (2.2) in the previous proof belongs to H^{p}.

Lemma. If 0 < p ≤ ∞ and F ∈ H^{p} ∩C^{ω}(ℝ), then there is a finite positive measure ν supported exactly on {F = 0} ∩ ℝ such that

F · Sν ∈ H^{p} ∩ C^{ω}(ℝ), Sν(z) := ∫ \frac{dν(t)}{t − z}.

Proof: Let {b_{k}} be all real zeros of F; we assume for simplicity that the zeros are simple. Choose small positive numbers ϵ_{k},

∑ ϵ_{k} < 1,  (2.4)

such that the ϵ_{k}-neighborhoods of b_{k} are disjoint. We have

|F (z)| ≤ C_{k}|z − b_{k}|

if z is in the ϵ_{k}-neighborhood of b_{k}. Take

ν = ∑ ν_{k}δ_{b_{k}}, ν_{k} = C_{k}^{−1}2^{−k}ϵ_{k},

and observe that

|Sν(z)| ≤ ∑_{k} \frac{ν_{k}}{|z − b_{k}|}.

If z is outside of all neighborhoods, then

|F Sν|(z) ≤ |F (z)| ∑ 2^{−k} = |F (z)|.

If z is in the k-th neighborhood, then

|F Sν|(z) ≤ |F (z)| + \frac{|F (z)|ν_{k}}{|z − b_{k}|} ≤ |F (z)| + C_{k}ν_{k} ≤ |F (z)| + 1.

We have F Sν ∈ H^{p} by (2.4). □

2.7. Kolmogorov-type criterion. The basic criterion is particularly useful in the case when we can represent γ as the argument of some explicitly given outer function, e.g., when U is the ratio of two inner functions with known de Branges functions.

Proposition. Let U = H̄/H, and H is an outer function real analytic and zero free on ℝ. Suppose dim N^{p}[U] < ∞. Then

N^{p}[U] ≠ 0 ⇔ H ∈ H^{p}.

Proof: If N^{p}[U] ≠ 0, then dim N^{p}[b^{s}U] = 1 for some s ≥ 0. We have

b^{s}U = H̄_{s}/H_{s}, H_{s} = (1 − b)^{−s}H.

By Proposition 2.6 we have a representation

H̄_{s}/H_{s} = F̄/F

for some outer function F ∈ H^{p} ∩ C^{ω}(ℝ), F ≠ 0 on ℝ. Since the function H_{s} is also outer and zero-free, it follows that H = const(1 − b)^{s}F and so H ∈ H^{p}. The converse is trivial. □

Comments. (a) If p = 2 and 1/H ∈ H^{2}, then the proof shows that the conclusion is true even without the assumption H ∈ C^{ω}(ℝ). The corresponding statement in the unit disc is equivalent to Kolmogorov’s minimality criterion in the theory of stationary Gaussian processes: {z^{n}} is minimal in L^{2}(w) iff w^{−1} ∈ L^{1}.

(b) The condition dim N^{p}[u] < ∞ is essential and related to the concept of ”rigid functions”, see [37]. We already mentioned two sufficient conditions for dim N^{p}[u] < ∞ in Section 2.3.

Example. This simple example is meant to illustrate the above criterion. Let Θ_{D} and Θ_{N} be the Weyl inner functions (1.7) for potential q = 0 on [0,1] with Dirichlet and Neumann boundary conditions at 0. We claim that the ”fractional dimension” of the kernel N[Θ̄_{N}Θ_{D}] is −1/4, see Section 2.1.

Proof: We represent the ratio Θ_{D}/Θ_{N} in terms of de Branges functions:

\frac{Θ_{D}}{Θ_{N}} = \frac{H̄}{H} , H = \frac{E_{D}}{E_{N}} ,

where

E_{N}(λ) = cos \sqrt{λ} − i\sqrt{λ} sin \sqrt{λ}, E_{D}(λ) = \begin{matrix}sin & \sqrt{λ} \\ √ & \overline{λ} \end{matrix} + i cos \sqrt{λ},

so

U := \frac{H̄}{H} , H = \frac{1}{\sqrt{λ}} · \frac{sin \sqrt{λ} + i\sqrt{λ} cos \sqrt{λ}}{cos \sqrt{λ} − i\sqrt{λ} sin \sqrt{λ}}.

Clearly, dim N[U] < ∞, and by Kolmogorov’s criterion we have N[b̄^{s}U] ≠ 0 iff F := (1 − b)^{s}H ∈ L^{2}(ℝ). If x > 0, then

|H(x)|^{2} = \frac{1}{x} \frac{sin^{2} \sqrt{x} + x cos^{2} \sqrt{x}}{cos^{2} \sqrt{x} + x sin^{2} \sqrt{x}}.

Let us estimate |F |^{2} on an interval I_{n} about π^{2}n^{2} where | sin \sqrt{x}| ≪ 1. If we write x = π^{2}n^{2} + s, so

|\sqrt{x} − πn| ≍ \frac{|s|}{n},  sin^{2} \sqrt{x} ≍ \frac{s^{2}}{n^{2}} ,  x sin^{2} \sqrt{x} ≍ s^{2},

then

|H(x)|^{2} ≍ \frac{1}{x} \frac{x}{1 + x sin^{2} \sqrt{x}} ≍ \frac{1}{1 + s^{2}} ,

|F (x)|^{2} ≍ n^{−4s} \frac{1}{1 + s^{2}} , ∫_{I_{n}} |F |^{2} ≍ n^{−4s}.

Thus F ∈ L^{2}(ℝ_{+}) iff 4s > 1. Finally we note that

|H(x)| ∼ \frac{1}{\sqrt{|x|}}, x → −∞,

and so F ∈ L^{2}(ℝ_{−}) iff s > 0. □

Similarly one can show that ” dim ”N[Θ̄_{D}Θ_{N}] = −5/4. One can also compute the dimensions of kernels in other Hardy spaces. In particular, ” dim ”N^{∞}[Θ̄_{N}Θ_{D}] = 0 and ” dim ”N^{∞}[Θ̄_{D}Θ_{N}] = −1; moreover, N^{∞}[Θ̄_{N}Θ_{D}] ≠ 0 and N^{∞}[b̄Θ̄_{D}Θ_{N}] ≠ 0.

2.8. Twin inner functions. We say that two meromorphic inner functions are twins if they have the same point spectrum (possibly including infinity). Twin functions appear in several applications (see, e.g., Section 3.5 below), and the main result is that the Toeplitz operator corresponding to their ratio is injective. This fact is quite different from Levinson’s type conditions discussed later in this section.

Theorem. Let Θ and J be twin meromorphic inner functions. Then N[Θ̄J] = 0.

Proof: We have

Θ̄J = \frac{H̄}{H} , H = \frac{1 − Θ}{1 − J}.

Since H^{±1} ∈ C^{ω}(ℝ), we can apply Kolmogorov’s criterion. (The kernel is finite dimensional because the argument of Θ̄J is bounded.) We claim that H ∉ H^{2}. Indeed, if Θ has no point mass at infinity, then

|H| ≥ \frac{1}{2}|1 − Θ| ∉ L^{2}.

If both functions have a point mass at infinity, then by l’Hôpital’s rule the angular limit

lim_{∞} \frac{1 − Θ}{1 − J} = \frac{Θ′(∞)}{J′(∞)}

exists and is non-zero (see Section 1.2), so H can not be in H^{2}. □

Remarks. (a) The proof shows that N[Θ̄J] = 0 if we have {Θ = 1} = {J = 1} and ∞ ∉ σ(Θ). Moreover, it can be shown that

σ(Θ) ⊂ σ(J)  ⇒  N[Θ̄J] = 0.

(b) Similar technique applies to symbols of the form u = Θ̄JH̄/H where Θ and J are twin inner functions, and H is an outer function real-analytic and zero free on ℝ. If dim N^{p}[H̄/H] < ∞, then

N^{p}[b̄^{s}Θ̄JH̄/H] ≠ 0 ⇔ (1 − b)^{s} \frac{1 − Θ}{1 − J}H ∈ H^{p}.

General form of Levinson’s completeness theorem

As we mentioned in Section 2.1, if ∃ γ(±∞), then

δ = γ(+∞) − γ(−∞) > −\frac{2π}{p}  ⇒  N^{p}[e^{iγ}] = 0,  (0 < p < ∞).

We extend this fact to general symbols by considering the ”mean” behavior at +∞ of the function

δ(x) = γ(x) − γ(−x).

Our result has the same form as Kolmogorov’s type condition in Section 3.7, but this time we don’t assume a priory that the kernel is finite dimensional. The new idea is to apply the Poisson A-integral transform to the equation (2.3) in the basic criterion. (See Section 3.4 for another application of this idea.)

2.9. Titchmarsh and Uly’anov Theorems. Let h ∈ L_{loc}^{1}(ℝ) be a real valued function. For each A > 0 we denote

h^{A} = \begin{cases}h(x), |h(x)| ≤ A, \\ 0, |h(x)| ≥ A. \end{cases}

The Schwarz A-integral of h is defined by the formula

S_{(A)}h(z) = lim_{A→∞} Sf^{A}(z), z ∈ C_{+},

provided that the limit exists for all z. Similarly, one defined the Poisson and the conjugate Poisson A-integrals P_{(A)}h and Q_{(A)}h respectively so that

S_{(A)}h = P_{(A)}h + iQ_{(A)}h.

Recall that if h, h̃ ∈ L_{Π}^{1}, then Sh̃ = −iSh + iSh(i).

Theorem. If h ∈ L_{Π}^{1}, then the Schwarz A-integral of h̃ exists, and we have

S_{(A)}h̃(z) = −iSh(z) + iSh(i),  z ∈ C_{+}.  (2.5)

The real part of the equation (2.5), or rather its special case

P_{(A)}h̃(i) = 0,  (2.6)

is due to Titchmarsh, see [42], and the imaginary part of (2.5),

Q_{(A)}h̃ = −P h + P h(i),  (2.7)

is the Uly’anov’s theorem, see [2] for a shorter proof. Note that we use a slightly different definition of the A-transforms but the definitions are in fact equivalent because h̃ ∈ L_{Π}^{o(1,∞)}.

2.10. A sufficient condition for N^{p}[U] = 0. For an odd function δ : ℝ → ℝ we define

Lδ(y) = \frac{2}{π} ∫_{0}^{∞} [\frac{1}{1 + t^{2}} − \frac{1}{y^{2} + t^{2}} ] tδ(t) dt, y > 0.

The integral makes sense (it might be + or - ∞) if we assume that

either ∫ ^{∞} δ^{+}(t) t^{−3}dt < ∞  or ∫ ^{∞} δ^{−}(t) t^{−3}dt < ∞,  (2.8)

where, as usual, δ^{±} := max{±δ, 0}. Note that if δ ∈ L_{Π}^{1}, then Lδ(y) = δ̃(iy).

Theorem. Let γ ∈ C^{ω}(ℝ) and let δ(x) = γ(x) − γ(−x) satisfy (2.8). Then

N^{p}[e^{iγ}] ≠ 0  ⇒  ∃F ∈ H^{p/4}(C_{+}), e^{Lδ(y)} ≤ |F (iy)|, (y > 1).

Proof: By (2.3) we have

γ = −ϕ + h̃_{1}, h_{1} ∈ L_{Π}^{1}, e^{−h_{1}} ∈ L^{p/2}.

Then

−γ(−x) = ϕ(−x) + h̃_{2}(x), h_{2} ∈ L_{Π}^{1}, e^{−h_{2}} ∈ L^{p/2}.

It follows that

δ = −ψ + h̃, h ∈ L_{Π}^{1}, e^{−h} ∈ L^{p/4},

where ψ is an odd increasing function. We apply the Uly’anov theorem to

h̃ = δ_{1} := ψ + δ.

By (2.7) we have

Q_{A}δ_{1} = Q_{A}h̃ = −Ph + const,

so

e^{Q_{A}δ_{1}} = const |F |, F := e^{−Sh} ∈ H^{p/4}

and it remains to show that

Lδ(y) ≤ Q_{A}δ_{1}(iy).

Since tδ_{1}(t) ≥ tδ(t) for all t ∈ ℝ, and since the kernel of L is positive for y > 1, we have

πQ_{A}δ_{1}(iy) = lim_{A→∞} ∫_{\{|δ_{1}|<A\}} [\frac{1}{1 + t^{2}} − \frac{1}{y^{2} + t^{2}} ] tδ_{1}(t) dt

≥ lim_{A→∞} ∫_{\{|δ_{1}|<A\}} [\frac{1}{1 + t^{2}} − \frac{1}{y^{2} + t^{2}} ] tδ(t) dt

= ∫_{ℝ} [\frac{1}{1 + t^{2}} − \frac{1}{y^{2} + t^{2}} ] tδ(t) dt = πLδ(y).

□

Remark. The proof works for any γ such that

γ = −ϕ + h̃, h ∈ L_{Π}^{1}, e^{−h} ∈ L^{p/2}.

We don’t need to assume γ ∈ C^{ω} as long as we have such a representation.

2.11. Levinson-type conditions. We can use standard growth estimates of Hardy space functions to derive more familiar looking conditions.

(a)

N^{p}[e^{iγ}] ≠ 0  ⇒  e^{Lδ(y)} = o (y^{−4/p}) ,  y → +∞,

In other words,

lim sup_{y→∞} [Lδ(y) + \frac{4}{p} log y] > −∞  ⇒  N^{p}[u] = 0.  (2.9)

Example. A simple computation shows

[Lsign](y) = \frac{2}{π} log y.

It follows that N[e^{iγ}] = 0 if γ(x) ≥ γ(−x) − π for x ≥ x_{0}. Indeed, we have δ ≥ −π for large x, and therefore

Lδ(y) ≥ −π[Lsign](y) + O(1) = −2 log y + O(1).

(b)

N^{p}[e^{iγ}] ≠ 0  ⇒  ∫ ^{∞} e^{\frac{p}{4} Lδ(y)}dy < ∞.

For example,

Lδ(y) ≥ −\frac{4}{p} log y − log log y  ⇒  N^{p}[e^{iγ}] = 0.

A more general statement is

N^{p}[e^{iγ}] ≠ 0  ⇒  ∫_{0}^{∞} e^{\frac{p}{4} Lδ(y)}dν(y) < ∞

for any Carleson measure ν in C_{+}.

(c) For each y > 1, the quantity

\frac{Lδ(y)}{[Lsign](y)} = \frac{π}{2} \frac{Lδ(y)}{log y}

is some sort of ”averaging” of δ near +∞. The meaning of (2.9) is that the ”mean” value of δ in infinity has to be ”less” than −2π/p for the Toeplitz kernel N^{p}[e^{iγ}] to be non-trivial. Here is a different way to express the same idea.

Corollary. Suppose δ(x) satisfies the integrability conditions (2.8). Then N^{p}[e^{iγ}] = 0 if

∫_{0}^{x} \frac{δ(t)}{t}dt ≥ −\frac{2π}{p} log x + O(1), x → +∞.

Proof: Integration by parts shows: if

∫_{0}^{x} \frac{a(t)}{t}dt ≤ log x + O(1), x → ∞,

then

∫_{0}^{∞} \frac{y^{2}}{y^{2} + t^{2}} · \frac{a(t)dt}{t} ≤ log y + O(1), y → ∞.

(If the latter integral is not converging, then we understand it as lim sup ∫_{0}^{x}.) □

3. Some applications

Completeness and minimality problem

3.1. Restatement in terms of uniqueness sets and Toeplitz kernels. We will study the following situation. Consider the Schrödinger equation (0.2) on an interval (a, b) with potential q and some fixed selfadjoint boundary condition β at b. We assume that the endpoint a is regular. For each λ ∈ C we choose a nontrivial solution u_{λ} satisfying the boundary condition; this solution is unique up to a constant. If Λ ⊂ C, we say that the family {u_{λ}}_{λ∈Λ} is complete if

span{u_{λ} : λ ∈ Λ} = L^{2}(a, b),

and is minimal if

∀λ_{0} ∈ Λ, u_{λ_{0}} ∉ span{u_{λ} : λ ∈ Λ \ λ_{0}}.

We will use the notation of Sections 1.4–1.6: Θ = Θ_{b,β}^{a} is the Weyl inner function, and E is a de Branges function of Θ. For λ ∈ C_{+} ∪ ℝ we have reproducing kernels k_{λ} and dual reproducing kernels k_{λ}^{∗} of K_{Θ}, see (1.10). For λ ∈ C, let K_{λ}^{E} denote the reproducing kernel in B(E). Finally, we represent

Λ = Λ_{+} ∪ Λ_{−}, Λ_{+} := Λ ∩ (C_{+} ∪ ℝ), Λ_{−} := Λ ∩ C_{−}.

Lemma. The following assertions are equivalent:

(i) the family {u_{λ}}_{λ∈Λ} is complete (minimal) in L^{2}(a, b);

(ii) the family {KE ; λ̄}_{λ∈Λ} is complete (minimal) in B(E);

(iii) the family {kλ̄}_{λ∈Λ_{−}} ∪ {k_{λ}^{∗}}_{λ∈Λ_{+}} is complete (minimal) in K_{Θ}.

Proof: This follows from Theorem 1.5 and and Corollary 1.7. □

We say that Λ ⊂ C is a uniqueness set of B(E) if there is no non-trivial function F ∈ B(E) such that F = 0 on Λ. The equivalence (i)⇔(ii) in the above lemma means that {u_{λ}}_{λ∈Λ} is complete if and only if Λ is a uniqueness set of B(E), and that {u_{λ}}_{λ∈Λ} is minimal if and only if Λ\λ_{0} is not a uniqueness set for any λ_{0} ∈ Λ.

We define uniqueness sets Λ of K_{Θ} in a similar way; in this case Λ ⊂ C_{+} ∪ ℝ. The definition obviously extends to divisors, i.e. sets of points with assigned multiplicities.

Lemma. Let Λ ⊂ C_{+} ∪ ℝ and let M ⊂ C_{+}. Then the family

{k_{λ}}_{λ∈Λ} ∪ {k_{μ}^{∗}}_{μ∈M}

is complete in K_{Θ} if and only if Λ ∪ M is a uniqueness divisor.

Proof: Suppose the family is not complete, so there is a non-trivial F ∈ K_{Θ} orthogonal to all k_{λ} and k_{μ}^{∗}. Let H ∈ K_{Θ} be defined by the relation

Θ̄F = H̄ on ℝ.

Then we have F = 0 on Λ and H = 0 on M. (Recall that Θ̄k_{μ} = k_{μ}^{∗}.) From the latter fact we have a representation

H = B_{M}G, G ∈ H^{2},

where B_{M} is the Blaschke product, and therefore

Θ̄(B_{M}F) = Ḡ.

The function B_{M}F is in K_{Θ} and is zero on Λ∪M. The opposite direction is similar. □

The equivalence (i) ⇔ (iii) in the first lemma now means that {u_{λ}}_{λ∈Λ} is complete in L^{2}(a, b) if and only if Λ_{+} ∪ \overline{Λ_{−}} is a uniqueness divisor of B(E). The characterization of minimality is similar.

It is very easy to characterize uniqueness sets (or divisors) Λ of K_{Θ} in the terms of Toeplitz kernels in the case Λ ⊂ C_{+}. (The case Λ ⊂ ℝ is discussed in the next subsection.) A necessary and sufficient condition for uniqueness is the following:

N[Θ̄B_{Λ}] = 0.

Indeed, f ∈ K_{Θ} is zero on Λ if and only if g = B̄_{Λ}f ∈ N[Θ̄B_{Λ}].

Let us summarize the above discussion.

Theorem. Let Λ = Λ_{+} ∪ Λ_{−}, Λ_{±} ⊂ C_{±}, and let B denote the Blaschke product corresponding to the divisor Λ_{+} ∪ \overline{Λ_{−}}. The family \{u_{λ}\}_{λ∈Λ} is complete in L^{2}(a, b) iff N[Θ̄B] = 0 and is minimal iff N[b̄Θ̄B] ≠ 0. The family is complete and minimal if and only if

dim [b̄Θ̄B] = 1.

The ”dimension” of the kernel, which may be negative, see Section 2.1, can be interpreted as the excess/deficiency of the family.

3.2. Uniqueness sets of K_{Θ}^{p}. We will consider the case of an arbitrary p > 0. As we explained, if Λ ⊂ C_{+}, then

Λ is a uniqueness set of K_{Θ}^{p}  ⇔  N^{p}[Θ̄B] = 0.  (3.1)

Now we concentrate on the case Λ ⊂ ℝ.

Proposition. Let Θ be a meromorphic inner function and Λ ⊂ ℝ. Then Λ is a uniqueness set of K_{Θ}^{p} if and only if N^{p}[Θ̄J] = 0 for every meromorphic inner function J such that {J = 1} = Λ.

Proof: Suppose we have a non-trivial function F ∈ K_{Θ}^{p} which is zero on Λ. By Lemma 2.6, we can find an inner function J with {J = 1} = Λ such that

G = \frac{F}{1 − J} ∈ H^{p}.

Then G ∈ K_{Θ}^{p}, and we have

Θ̄JG = Θ̄(G − F) = Θ̄F \frac{J}{1 − J} = −\frac{Θ̄F}{1 − J̄} ∈ H̄^{p},

so the Toeplitz kernel is non-trivial.

Conversely, if G is a non-trivial element of N^{p}[Θ̄J], then F = (1 − J)G ∈ K_{Θ}^{p}, and so Λ is not a uniqueness set. Indeed, since G ∈ N^{p}[Θ̄J], we have JG ∈ K_{Θ}^{p}, and therefore G ∈ K_{Θ}^{p} and G − JG ∈ K_{Θ}^{p}. □

Compared to the case Λ ⊂ C_{+}, the condition in the last lemma seems to be less useful since it involves an infinite set of inner functions J. Nevertheless, we can overcome this difficulty and restate the criterion in both cases Λ ⊂ C_{+} and Λ ⊂ ℝ in a unified way.

If Λ ⊂ C_{+}, then combining (3.1) and (2.3) we see that Λ is not a uniqueness set of K_{Θ}^{p} iff the function

γ = arg B_{Λ} − arg Θ  (3.2)

has a representation

γ = −ϕ + h̃, h ∈ L_{Π}^{1}, e^{−h} ∈ L^{p/2},

where ϕ is the argument of a meromorphic inner function.

In the case Λ ⊂ ℝ, we will use the counting function n_{Λ} of Λ instead of the argument function of B_{Λ} in (3.2). By definition,

n_{Λ} = ∑ n_{λ},  n_{λ} = \begin{cases}χ(λ, +∞), λ > 0, \\ −χ(−∞, λ), λ < 0. \end{cases}  (3.3)

Theorem. Let Θ be a meromorphic inner function and Λ ⊂ ℝ. Then Λ is not a uniqueness set of K_{Θ}^{p} if and only if the function

γ = 2πn_{Λ} − arg Θ

has a representation

γ = −ϕ + h̃,  h ∈ L_{Π}^{1}, e^{−h} ∈ L^{p/2},  (3.4)

where ϕ is the argument of a meromorphic inner function.

Proof: If Λ is not a uniqueness set, then by the last proposition there is a meromorphic inner function J such that {J = 1} = Λ and such that the kernel N^{p}[Θ̄J] contains an outer function G which has no zeros ℝ. We have

Θ̄J = −Φ̄\frac{Ḡ}{G},

for some inner function Φ, and if we denote F = (1 − J)G, then

Θ = Φ\frac{F}{F̄} = Φ exp \{2i(log |F |)^{∼}\} = Φ exp \{i [2(log |F |)^{∼} + 2πn_{Λ}]\}.

The key observation is that the function in the square brackets is continuous – this follows from the fact that F is an outer function with zero set Λ, and from the identity (log |x|)^{∼} = πχ_{ℝ_{−}}. We conclude that

γ = −ϕ + h̃, h := −2 log |F |,

which proves one half of the statement. Reversing the argument we get the second half. □

Example. Λ is a uniqueness set of K^{p}[S^{a}] if γ(t) = 2πn_{Λ}(t) − at has a representation (3.4). As in Section 2.10 consider the function

δ(t) = γ(t) − γ(−t) = 2πN(t) − 2at, N(t) = #[Λ ∩ (−t, t)].

Applying Corollary 2.11 (and also Remark 2.10) we get a sufficient condition

∫_{1}^{x} \frac{N(t)}{t}dt ≥ \frac{a}{π} x − \frac{1}{p} log x + O(1), (x → +∞).

In fact, Theorem 2.10 implies the latter condition in the lim sup sense, as well as some other sufficient conditions. Similar results can be stated for various families of special functions.

Zero sets of entire functions

3.3. Exact zero sets of B^{+}(E)-functions. We will be considering general Cartwright-de Branges spaces, see Section 1.7. A set Λ ⊂ C is said to be an exact zero set of B^{+}(E) if there exists an entire function F ∈ B^{+}(E) such that F = 0 exactly on Λ. We’ll restrict the discussion to the case Λ ⊂ ℝ.

Recall that B^{+}(E) = EK^{+}(Θ) where Θ = Θ_{E} = E^{#}/E, so Λ ⊂ ℝ is also an exact zero set of K^{+}(Θ). The following description is essentially our basic criterion in Proposition 2.5. In the next subsection we will see that in the special case E = S^{−a}, this description contains main results of the theory of Cartwright’s functions. As usual, S denotes the singular inner function e^{iz}.

Theorem. Λ ⊂ ℝ is an exact zero set of B^{+}(E) if and only if

2πn_{Λ} − arg Θ_{E} = −bx + h̃, h ∈ L_{Π}^{1}, b ≥ 0.

Proof: We need to show that a function F ∈ B^{+}(E) with zero set Λ exists if and only

Θ = JS^{b} \frac{H}{H̄}  (3.5)

for some inner function J with {J = 1} = Λ, some b ≥ 0, and some outer function H ∈ C^{ω}(ℝ) which has no zeros on ℝ.

⇐: The function

F (z) = \begin{cases}(1 − J(z)) H(z)E(z)S^{b}(z),  z ∈ C_{+} \\ (J^{#}(z) − 1) H^{#}(z)E^{#}(z),  z ∈ C_{−} \end{cases}

is in B^{+}(E) and vanishes exactly on Λ.

⇒: Suppose F ∈ B^{+}(E) vanishes exactly on Λ. Since B^{+}(E) = EK_{Θ}^{+}, there is G ∈ K_{Θ}^{+} such that F = EG in C_{+}. Clearly, we can assume that G is an outer function. Define G_{−} ∈ N^{+}(C_{+}) by the equation

Θ̄G = Ḡ_{−} on ℝ.

Since

F (z) = E^{#}(z)\overline{G_{−}(z̄)}, z ∈ C_{−},

G_{−} has no zeros in C_{+}, and therefore its inner-outer factorization has the form

G_{−} = S^{b}G,  b ≥ 0.  (3.6)

We now take any meromorphic inner function J with {J = 1} = Λ and define

H = \frac{G}{1 − J} .

This is an outer function with the stated properties. We have

G = H(1 − J), G_{−} = H(1 − J)S^{b},

and from (3.6) we get the representation (3.5). □

Example: Λ ⊂ ℝ is an exact zero set of a Cartwright function if and only

n_{Λ}(x) = cx + h̃,  c ≥ 0, , h ∈ L_{Π}^{1}.  (3.7)

3.4. Zeros of Cartwright’s functions. We need the following elementary lemma.

Lemma. Suppose g ∈ L_{Π}^{o(1,∞)} and g′ ≥ −const. Then

g(x) = o(x), x → ±∞.

Proof: If g(x_{∗}) ≥ cx_{∗}, x_{∗} ≫ 1, then for x > x_{∗} we have (assuming g′ ≥ −1)

g(x) ≥ g(x_{∗}) − (x − x_{∗}) ≥ (1 + c)x_{∗} − x,

and so g(x) ≳ x_{∗} on an interval [x_{∗}, (1+δ)x_{∗}]. The Poisson measure of this interval is ≍ 1/x_{∗}, which contradicts the weak L^{1}-condition. □

Corollary. Let Λ ⊂ ℝ be the exact zero set of some Cartwright function. Then

∃c ≥ 0,  lim_{x→−∞} \frac{n_{Λ}(x)}{x} = lim_{x→+∞} \frac{n_{Λ}(x)}{x} = c.

Proof: From (3.7) we conclude that g(x) = n_{Λ}(x) − cx satisfied the conditions of the lemma. □

Corollary. If Λ ⊂ ℝ is the exact zero set of a Cartwright function, then there exists a limit

v.p. ∑_{λ∈Λ} \frac{1}{λ}.

Proof: For every B > 0 we define the function

η_{B}(t) = \begin{cases}B, t > B, \\ t, t ∈ [−B, B], \\ −B, t < −B. \end{cases}

Using the notation (3.3), we have

∑_{|λ|≤B} n_{λ} = n_{Λ} ∘ η_{B}.

We also introduce an elementary function

g(λ) = ∫ n_{λ}dΠ = \frac{π}{2} sign(λ) − arctan λ = \frac{1}{λ} + O (\frac{1}{λ^{2}} ) , λ → ±∞.

By the previous corollary we have

∃ v.p. ∑ \frac{1}{λ}  ⇔  ∃ v.p. ∑ g(λ).

By (3.7), n_{Λ} = cx + f, where we write f for h̃, and therefore

∑_{|λ|≤B} g(λ) = ∫ n_{Λ} ∘ η_{B} dΠ = ∫ f ∘ η_{B} dΠ + const.

The only properties of f that we use in the rest of the proof are: f(x) = o(x) as x → ∞, and Titchmarsh’s theorem (2.6),

lim_{A→∞} ∫ f^{A}dΠ = 0, f^{A} = f · (χ_{[−A,A]} ∘ f).

Since f(x) = o(x), we have

sup_{[−B,B]} |f| = o(B), B → ∞.

It follows that there is a function A = A(B) such that A = o(B), A(∞) = ∞, and

f = f^{A(B)} on [−B, B].

Then we also have

|f ∘ η_{B}|, |f^{A(B)}| ≤ A(B) on ℝ,

and therefore

∫ |f ∘ η_{B} − f^{A(B)}| dΠ ≤ A(B) ∫_{ℝ\[−B,B]} dΠ → 0, B → ∞.
□

We leave it to the reader to state relevant results concerning zeros of functions from more general Cartwright-de Branges spaces B^{+}(E). For example, the key property h̃ = o(t) is valid if

lim_{δ→0} lim sup_{x→∞} \frac{θ(x + δx) − θ(x)}{x} = 0,

where θ is the argument of Θ_{E}.

Spectral problems with mixed data

3.5. Abstract Hochstadt-Liberman problem. We will be considering the following problem concerning general meromorphic inner functions. In the next section we will explain its relation to Hochstadt-Liberman’s theorem [21].

Let Φ and Ψ be meromorphic inner function and Θ = ΨΦ. As usual, σ(Θ) denotes the (point) spectrum of Θ, see Section 1.2; recall that σ(Θ) may include ∞. We say that the data [Ψ, σ(Θ)] determine Θ if for any inner function Φ̃,

Θ̃ = ΨΦ̃, σ(Θ̃) = σ(Θ)  ⇒  Θ = Θ̃.

Alternatively, we can say that Ψ and σ(ΦΨ) determine Φ. Given Φ and Ψ, the problem is to decide if this is the case.

The set of Herglotz measures of inner functions Θ̃ satisfying Ψ|Θ̃ and σ(Θ̃) = σ(Θ) is convex, see Section 1.2. We will refer to the dimension of this set as the dimension of the set of solutions.

Example. Suppose Θ is a finite Blaschke product. Then

[Ψ, σ(Θ)] determine Θ  ⇔  2 deg Ψ > deg Θ.

The proof is elementary; it also follows from the results below. As an illustration consider the simplest case Θ = b^{2}, Ψ = b. Then σ(Θ) = {0, ∞}, and the data [Ψ, σ(Θ)] does not determine Θ. In fact, the set of solutions is one-dimensional; the solutions are given by the formula

Φ̃(z) = \frac{z − ia}{z + ia}, (a > 0).

We will state some conditions in terms of the Toeplitz kernels with symbol U = Φ̄Ψ. The rough meaning of these conditions is the following: for the data [Ψ, σ(Θ)] to determine Θ, the known factor Ψ of the inner function has to be ”bigger” than the unknown factor Φ.

Proposition. If N^{∞}[Φ̄Ψ] ≠ {0}, then the data [Ψ, σ(Θ)] does not determine Θ.

Proof: Take

a ∈ N^{∞}[Θ̄Ψ^{2}], ∥a∥_{∞} < \frac{1}{2}.

Then

Θ̄Ψ^{2}a = b̄,

and

Θ̄Ψ^{2}(a + b) = ā + b̄.

Denote

g = a + b, f = Ψg,

so that

Θ̄f = f̄, Ψ|f, ∥f∥_{∞} < 1.

Then the function

Θ̃ = \frac{f + Θ}{f + 1}  (3.8)

is inner:

|Θ̃|^{2} = \frac{(f + Θ)(f̄+ Θ̄)}{(f + 1)(f̄+ 1)} = \frac{(f + Θ)(Θ̄f + Θ̄)}{(f + 1)(Θ̄f + 1)} = 1,

and Ψ|Θ̃. Let us check that σ(Θ) = σ(Θ̃). This is clear for finite eigenvalues, and we also note that ∞ ∈ σ(Θ) iff ∞ ∈ σ(Θ̃) because Θ − 1 ∈ H^{2} iff

Θ̃ − 1 = \frac{Θ − 1}{f + 1} ∈ H^{2}.

□

Here is a partial converse. We write N_{Π}^{p}[U] for N^{+}[U] ∩ L_{Π}^{p}.

Proposition. If N_{Π}^{p}[Φ̄Ψ] = {0} for some p < 1, then [Ψ, σ(Θ)] determine Θ.

Proof: Suppose σ(Θ_{1}) = σ(Θ) and Ψ|Θ_{1}. Then the function

f = \frac{Θ_{1} − Θ}{1 − Θ_{1}}

is in H_{Π}^{p} ∩ C^{ω}(ℝ) for all p < 1 because (1 − Θ_{1})^{−1} has positive real part in C_{+} and Θ_{1} = Θ on {Θ_{1} = 1}. Observe that

Θ̄f = Θ̄\frac{Θ_{1} − Θ}{1 − Θ_{1}} = \frac{Θ_{1}Θ̄ − 1}{1 − Θ_{1}} = \frac{Θ̄ − Θ̄_{1}}{Θ̄_{1} − 1} = f̄.

Since Ψ|f, we can define

g = Ψ̄f ∈ H_{Π}^{p} ∩ C^{ω}(ℝ).

We have

Φ̄Ψg = Θ̄Ψ^{2}g = ΨΘ̄f = Ψf̄ = ḡ,

and so g ∈ N_{Π}^{p}[Φ̄Ψ] = {0} and Θ_{1} = Θ. □

How big is the gap between the conditions in the above statements? As we will see in Section 4.1, the gap is just finite dimensional if the inner function Φ is not very ”wild”. Namely, if the argument of Φ has a polynomial growth at infinity, then N_{Π}^{p}[Φ̄Ψ] = {0} implies N^{∞}[b^{n}Φ̄Ψ] = {0} for some n < ∞.

We now demonstrate a different way to state a partial converse of the first statement. We get precisely the converse statement up to a factor which is the ratio of two twin inner functions, see Section 2.8.

Proposition. If the data [Ψ, σ(Θ)] don’t determine Θ, then there are inner functions Θ_{1} and J such that {Θ_{1} = 1} = {J = 1} and

N^{∞}[(Θ̄_{1}J)Φ̄Ψ] ≠ 0.

Proof: Suppose we have σ(Θ_{1}) = σ(Θ), Θ_{1} = Φ_{1}Ψ. Then the function Φ − Φ_{1} is in K^{∞}[ΦΦ_{1}] and is zero on {Θ_{1} = 1}. By Proposition 3.2, there is an inner function J such that J = 1 exactly on {Θ_{1} = 1} and N^{∞}[Φ̄_{1}Φ̄J)] ≠ 0. Finally, we note

Φ̄Φ̄_{1}J = Φ̄Ψ(Θ̄_{1}J).

□

We can apply our results on Toeplitz kernels (Sections 2 and 4) to obtain various necessary or sufficient conditions in the Hochstadt-Liberman problem. Here is a simple example that extends the original Hochstadt-Liberman theorem, which we will recall in the next subsection.

Corollary. Suppose Θ = Ψ^{2} and ∞ ∉ σ(Θ). Then the set of solutions is exactly one-dimensional: Θ̃ satisfies Ψ|Θ̃, σ(Θ̃) = σ(Θ) iff

∃r ∈ (−1, 1),  Θ̃ = Ψ\frac{r + Ψ}{1 + rΨ}.  (3.9)

Proof: We have Φ = Ψ, so N^{∞}[Φ̄Ψ] ≠ 0 and by the first proposition the dimension is at least 1. On the other hand, the dimension can not be ≥ 2, for otherwise by the last proposition we would have N^{∞}[bΘ̄_{1}J] ≠ 0 and therefore N[Θ̄_{1}J] ≠ 0 for some inner functions Θ_{1} and J such that {J = 1} = {Θ_{1} = 1} and σ(Θ_{1}) = σ(Θ). By assumption, ∞ ∉ σ(Θ_{1}), so and we get a contradiction with the twin function theorem in Section 2.8. The formula (3.9) now follows from the construction (3.8) in the proof of the first proposition. □

Remark. One can show that the statement is true even without assumption ∞ ∉ σ(Θ). Also, one can state the following ”one-sided” version (see Remark (a) in Section 2.8):

Let Θ = Ψ^{2}. Then Θ̃ satisfies Ψ|Θ̃, σ(Θ̃) ⊂ σ(Θ) iff

∃r ∈ [−1, 1],  Θ̃ = Ψ\frac{r + Ψ}{1 + rΨ}.

In other words, the (convex) set of all such Θ̃’s is the segment [−Ψ, Ψ]. Once we know that the dimension is one, the formula follows from the obvious fact that ±Ψ are extreme points of this set.

3.6. Spectral theory interpretation: Hochstadt-Liberman and Khodakovski theorems. Consider a Schrödinger operator L = (q, α, β) on (a, b), where q ∈ L_{loc}^{1}(a, b) and α, β are selfadjoint boundary conditions at a and b respectively; the endpoints can be infinite and/or singular. We assume that L has compact resolvent. As usual, σ(L) denotes the spectrum of L.

Suppose a < c < b. We will write q_{−} for the restriction of q to (a, c) and q_{+} for the restriction of q to (c, b). We say that the data (q_{−}, α, σ(L)) determines L if for any other Schrödinger operator L̃ = (q̃, α̃, β̃),

q_{−} = q, α = α̃, σ(L̃) = σ(L)  ⇒  q̃_{+} = q_{+}, β̃ = β.

Let Θ_{−} denote the Weyl inner function of (q_{−}, α) computed at c and Θ_{+} the Weyl inner function of (q_{+}, β) computed at c.

Lemma. σ(L) = σ(Θ_{−}Θ_{+}).

Proof: The equation Θ_{−}(λ)Θ_{+}(λ) = 1 is equivalent to the statement

m_{+}(λ) + m_{−}(λ) = 0 or m_{−}(λ) = m_{+}(λ) = ∞

for the corresponding m-functions. The latter means that we have the matching

\frac{u̇_{−}(c, λ)}{u_{−}(c, λ)} = \frac{u̇_{+}(c, λ)}{u_{+}(c, λ)}

for any two non-trivial solutions u_{−}(·, λ) and u_{+}(·, λ) of the Schrödinger equation with boundary conditions α and β respectively, which is possible if and only if λ is an eigenvalue of L. □

Corollary. (q_{−}, α, σ(L)) determine L if the data (Θ_{−}, σ(Θ_{−}Θ_{+})) determine Θ_{+}.

Here we of course rely on the fundamental uniqueness theorem of Borg and Marchenko [Borg], [M1]: the m-function (and therefore the Weyl inner function) determines both the potential and the boundary condition.

Remark. We would have an ”iff” statement if we considered the problem in some class of canonical systems with a one-to-one correspondence between the systems and inner functions such as the class of Krein’s ”strings”, see [12], [14]. The effective characterization of inner functions of Schrödinger operators is an open problem, so we will use our general results to state only sufficient conditions for Schrödinger operators. To obtain necessary condition one has to use more specific techniques of the Schrödinger operator theory, see [7], [22].

Let us apply the above corollary to the situation described at the end of the last subsection.

Example 1. Let L be a Schrödinger operator on ℝ with compact resolvent and limit point boundary conditions at ±∞. Suppose the potential q(x) is an even function:

q(−x) = q(x), (x > 0).

Then q|_{ℝ_{−}} and σ(L) determine L.

Proof: By Everitt’s theorem [15] (we recall it in the next subsection), all the inner functions (r + Ψ)/(1 + r + Ψ) in (3.9) with r ≠ 0 are not Weyl inner functions corresponding to a Schrödinger operator. □

This result is a special case of Khodakovski’s theorem [24], where only the equality q(−x) ≤ q(x) for x > 0 is assumed. The full version of Khodakovski’s theorem requires a slightly different approach which we describe in the next subsection. Similarly, from the remark at the end of Section 2.5 we derive the following statement.

Let L be as above, and let L̃ be another Schrödinger operator on (−∞, b), b ≥ 0. If

q = q̃ on ℝ_{−}  and  σ(L̃) ⊂ σ(L),

then either L̃ = L or b = 0 and L̃ is the operator with potential q_{−} and Dirichlet or Neumann condition at 0.

Example 2. Let L be a regular selfadjoint Schrödinger operator on [a, b] with non-Dirichlet boundary conditions α and β at a and b respectively. If c = (a + b)/2, then (q_{−}, α, σ(L)) determine L.

The statement is also true if one or both boundary conditions are Dirichlet, see next subsection. This is a stronger version of the Hochstadt-Liberman theorem [21], see also [17] which states that if both L and L̃ are regular, and q̃_{−} = q_{−}, α̃ = α, σ(L̃) = σ(L), then L̃ = L. We do not require L̃ to be regular. Also, we can replace σ(L̃) = σ(L) with σ(L̃) ⊂ σ(L).

3.7. Everitt’s class of inner functions. We need the following well-known fact, see [15]:

If m(z) is an m-function of a Schrödinger operator, then

m(z) = i\sqrt{z} + o(1), (z → ∞, z ∈ iℝ_{+}).

It follows that if Ψ(z) is a Weyl inner function of a Schrödinger operator, then

Ψ(z) = 1 − \frac{2}{\sqrt{z}} + \frac{2}{z} + o (\frac{1}{z}) ,  (z → ∞, z ∈ iℝ_{+}).  (3.10)

This motivates the following definitions. We say that a meromorphic inner function Ψ belongs to the class (Ev) if it satisfies the asymptotic relation (3.10). (Note though that (3.10) is by no means a full characterization of Weyl inner functions of Schrödinger operators.)

Definition. Let Ψ, Φ ∈ (Ev). We say that [Ψ, σ(ΨΦ)] determine Φ in the class (Ev) if

Φ̃ ∈ (Ev), σ(ΨΦ) = σ(ΨΦ̃)  ⇒  Φ̃ = Φ.

Proposition. Let Ψ, Φ ∈ (Ev). Suppose

∃p < 1, dim N^{p}[Φ̄Ψ] < ∞.

Furthermore, suppose we have a representation Φ̄Ψ = H̄/H, where H is an outer function such that H^{±1} ∈ C^{ω}(ℝ), and

H ≠ o (\frac{1}{\sqrt{|z|}}) , (z → ∞, z ∈ iℝ_{+}).

Then (Ψ, σ(ΨΦ)) determine Φ in the class (Ev).

Proof: We first argue as in the proof of the second proposition in Section 3.5. The function

G = \frac{Φ̃ − Φ}{1 − Φ̃Ψ} ∈ H_{Π}^{p} ∩ C^{ω}(ℝ)

satisfies

Φ̄ΨG = Ḡ.  (3.11)

We also derive from (3.10) that

G(z) = o (\frac{1}{\sqrt{z}} ) , (z → ∞, z ∈ iℝ_{+}).

Since the dimension of the Toeplitz kernel is finite, G has at most finitely many zeros on ℝ and its inner factor is a finite Blaschke product (if any). Thus we can assume that G is an outer function zero free on ℝ satisfying (3.11). But in this case H = G, and we have a contradiction. □

(Remark. One can show that the statement is true for all p ≤ 2.)

Examples. (i) If Φ = Ψ ∈ (Ev), then (Ψ, σ(ΨΦ)) determine Φ in the class (Ev). Proof: H = 1 ≠ o(|z|^{−1/2}). □

(ii) If Ψ = Θ_{D} and Φ = Θ_{N}, see (1.7), then Ψ and σ(ΨΦ) determine Φ in (Ev), and if Ψ = Θ_{N} and Φ = Θ_{D}, then Ψ and the spectrum σ(ΨΦ) minus any one point determine (in an obvious sense) Φ in the class (Ev). In particular, we have the Hochstadt-Liberman type result for all regular operators with arbitrary boundary conditions.

Proof: Reasoning as in Example 2.7, in the first case we have H = E_{D}/E_{N} and so |H| ∼ |z|^{−1/2}. In the second case, |H| = |E_{N}/E_{D}| ∼ |z|^{1/2}. □

Example: Bessel inner functions. This is an extension of the previous example. We want to show that the Hochstadt-Liberman phenomenon occurs not only for regular potentials.

We consider the Bessel inner functions Θ_{ν}, ν ≥ −1/2, see (1.8). Recall that

E_{ν}(λ) = (1 + i/2 + iν)G_{ν}(\sqrt{λ}) + iF_{ν}(\sqrt{λ})

is a de Branges function of Θ_{ν}, where G_{ν}(z) = z^{−ν}J_{ν}(z) and F_{ν}(z) = zG′_{ν}(z), see Section 1.6. If Ψ = Θ_{ν_{1}} and Φ = Θ_{ν_{2}}, then we have a representation

Φ̄Ψ = \frac{H̄}{H} , H = \frac{E_{ν_{1}}}{E_{ν_{2}}} .

Lemma.

|\frac{E_{ν_{1}}(λ)}{E_{ν_{2}}(λ)}| ∼ |λ|^{\frac{ν_{2}−ν_{1}}{2}} , λ ∈ iℝ_{+}, λ → ∞.

Proof: It is known that |J_{ν}(z)| ∼ |J_{0}(z)| as z → ∞ in any Stolz angle in C_{+}. Therefore,

|G_{ν}(z)| ∼ |z^{−ν}J_{0}(z)|,

and we also have

|F_{ν}(z)| ∼ |G_{ν−1}(z)| ∼ |z^{1−ν}J_{0}(z)|.

(The first relation follows from the identity zJ′_{ν}(z) = zJ_{ν−1}(z) − νJ_{ν}(z), which implies F_{ν} = G_{ν−1} − 2νG_{ν}.) □

We can now apply Proposition. In particular, we get the following result.

Theorem. Let L be the Schrödinger operator with potential q(t) = 2t^{−2} on [0, 2] and with Dirichlet boundary condition at t = 2. Then q|_{(0,1)} and the spectrum σ(L) determine L in the class of Schrödinger operators.

Proof: In our usual notation we have

Ψ = Θ_{3/2},  Φ̄Θ_{1/2} = \frac{F̄}{F} , F^{±1} ∈ H^{∞},

see Section 2.4. Therefore,

Φ̄Ψ = (Φ̄Θ_{1/2})Θ̄_{1/2}Θ_{3/2}

= \frac{F̄}{F} \frac{E_{1/2}}{Ē_{1/2}} \frac{Ē_{3/2}}{E_{3/2}} = \frac{H̄}{H} ,

where

H = \frac{F E_{3/2}}{E_{1/2}}

is an outer function such that H^{±1} ∈ C^{ω}(ℝ) and H(iy) ≠ o(1/\sqrt{y}) as y → +∞. We can apply the proposition because the argument of the unimodular function Φ̄Ψ is bounded, which is a consequence of the well-known asymptotic for the zeros of Bessel’s functions. □

Defining sets

3.8. Defining sets of inner functions. Let Φ = e^{iϕ} be a meromorphic inner function, and let Λ ⊂ ℝ. We say that Λ is a defining set for Φ if

Φ̃ = e^{iϕ̃}, ϕ̃ = ϕ on Λ  ⇒  Φ ≡ Φ̃.

In this definition we tacitly assume ϕ(±∞) = ±∞. In the ”one-sided” case, say if ϕ(−∞) > −∞ and ϕ(+∞) = +∞, one should modify the definition in an obvious way.

One can extend this definition to divisors. For instance, if all points in Λ ⊂ σ(Φ) are double, then the equality Φ̃ = Φ on Λ means that the spectral measures the inner functions coincide on Λ.

Let us mention several special cases.

(a) Two spectra problem. This corresponds to the case

Λ = {Φ = 1} ∪ {Φ = −1}.

The meaning of the following statement is that Λ is defining for Φ with deficiency one (in the case ϕ(±∞) = ±∞, to be accurate). Various related statements are of course well-known, see e.g. [7].

Let Φ be a meromorphic inner function. Then a meromorphic inner function Φ̃ satisfies {Φ̃ = 1} = {Φ = 1} and {Φ̃ = −1} = {Φ = −1} iff

Φ̃ = \frac{Φ − c}{1 − cΦ},  c ∈ (−1, 1).  (3.12)

The easiest way to see this is use Krein’s shift construction: since

ℜ [\frac{1}{πi} log \frac{Φ̃ + 1}{Φ̃ − 1}] = χ_{e} on ℝ,

where e = {ℑΦ > 0}, we have

\frac{1}{πi} log \frac{Φ̃ + 1}{Φ̃ − 1} = Sχ_{e} + const. □

This argument also shows that given any two intertwining discrete sets Λ_{±} of real numbers there is a meromorphic inner function Φ such that

{Φ = ±1} = Λ_{±}.

Let us also mention that the statement (3.12) can be derived from the twin inner function theorem, see Section 2.8. For instance, suppose I := Φ̃Φ has no point mass at infinity. We want to show that Φ̃ − Φ = c(1 − I). It is enough to check that the dimension of the set of functions F ∈ K_{I}^{∞} vanishing on {I = 1} is at most one. (Obviously, both Φ̃ − Φ and 1 − I belong to this set.) If not, we would have dim N^{∞}[ĪJ] ≥ 2 for some J vanishing on {I = 1}. But then N^{∞}[bĪJ] ≠ 0, and N[ĪJ] ≠ 0, which is impossible by the twin inner function theorem. □

(b) General mixed data spectral problem. The Hochstadt-Liberman problem for inner functions that we discussed above can be viewed as a special case of the defining sets problem. It is easy to see that if (assuming arg Θ(±∞) = ±∞) Θ = ΨΦ and Λ = σ(Θ), then

(Ψ, σ(Θ)) determine Θ  ⇔  Λ is defining for Φ.

This can be generalized in the following way. Let Θ = ΨΦ be a given meromorphic inner function and let {λ_{n}} be the set of its eigenvalues numbered in the increasing order. Given M ⊂ Z we denote

σ_{M}(Θ) = {λ_{n} : n ∈ M}.

The question is whether the factor Ψ and the partial spectrum σ_{M}(Θ) determine Θ, i.e. whether

Θ̃ = ΨΦ̃, λ̃_{n} = λ_{n} (n ∈ M)  ⇒  Θ̃ ≡ Θ.

Once again, this is equivalent (assuming ϕ(±∞) = ±∞) to saying that Λ = σ_{M}(Θ) is a defining set for Φ. The spectral theory meaning was explained in Section 3.6 (but now we consider eigenvalues from different spectra), and the partial spectral problem for Schrödinger operators and Jacobi matrices appeared in several publications, e.g. [17], [18].

(c) A version for spectral measures. Given a meromorphic inner function Θ and a factor Ψ|Θ, and also given a part of the spectrum Λ = σ_{M}(Θ), the question is whether there is another inner function Θ̃ ≠ Θ such that Ψ|Θ̃ and the spectral measures μ = μ_{Θ} and μ̃ = μΘ̃ coincide on Λ:

λ̃_{n} = λ_{n}, μ̃{λ_{n}} = μ̃{λ_{n}}, (n ∈ M).

Claim: If Θ = ΨΦ, then Ψ and the spectral measure on Λ = σ_{M}(Θ) determine Θ iff the divisor 2χ_{Λ} is defining for Φ. Indeed, if Θ̃ = ΨΦ̃, and

arg Θ̃(λ_{n}) = arg Θ̃(λ_{n}) = 2πn, (n ∈ M),

then

arg Φ̃ = arg Φ on Λ.

The relation

μ{λ} = μ̃{λ}, λ ∈ Λ

then implies, see (1.4), Θ̃′(λ) = Θ′(λ), so

Ψ′(λ)Φ̃(λ) + Ψ(λ)Φ̃′(λ) = Ψ′(λ)Φ(λ) + Ψ(λ)Φ′(λ), (λ ∈ Λ),

and

(arg Φ̃)′ = (arg Φ)′  on Λ.  □

Again, the spectral theory interpretation is the same as above: we know some part of a differential operator and some part of its spectral measure and we want to know if this information determines the operator uniquely.

As usual we can consider the problem in a restricted class of inner functions. Here is the simplest example.

Example. Let Θ = ΨΦ be a finite Blaschke product. Then Ψ and Λ ⊂ σ(Θ) determine Θ iff #Λ > 2 deg Φ in the class of Blaschke products of a fixed degree. Similarly, Ψ and the spectral measure on Λ determine Θ iff #Λ > deg Φ. This extends in an obvious way to the cases where only Φ or Ψ has a finite degree. These facts follow for instance from the statements in the next section, also cf. [18].

3.9. Relation to uniqueness sets.

Proposition. Λ is not defining for Φ if there is a non-constant function G ∈ K_{Φ}^{∞} such that

G = Ḡ on Λ.  (3.13)

Proof: We can assume ∥G∥_{∞} < 1. Define F ∈ H^{∞} by the equation Φ̄G = F̄ on ℝ, and consider

Φ̃ = \frac{Φ + F}{1 + G} .

Then Φ̃ is an inner function because it is in N^{+} and

|Φ + F | = |Φ + ΦḠ| = |1 + G| on ℝ.

Also, Φ̃ ≠ Φ because otherwise we would have F = GΦ, which together with F = ΦḠ implies G = Ḡ, so G = const. Finally, we have

Φ̃ = Φ\frac{1 + Φ̄F}{1 + G} = Φ\frac{1 + Ḡ}{1 + G} = Φ on Λ,

and since

∥ arg Φ̃ − arg Φ∥_{L^{∞}(ℝ)} < 2π

by construction, we get arg Φ̃ = arg Φ on Λ. □

Remark. The condition (3.13) is very close to the condition that Λ is a not a uniqueness set for K^{∞}[Φ^{2}]. The precise relation between the two statements is an interesting question, which we will not discuss here. We only mention that if p ∈ (1, ∞), then

∃G ∈ K^{p}[Φ], G ≢ const, G = Ḡ on Λ,

iff

∃F ∈ K^{p}[Φ^{2}], F ≢ 0, F = 0 on Λ.

The above proposition gives a necessary condition for a set Λ to be defining for Φ. To get sufficient conditions one can use the following obvious observation.

Lemma. If Φ̃ = Φ on Λ and F = Φ̃ − Φ, then

F ∈ K^{∞}[Φ̃Φ], F = 0 on Λ.

If we also have arg Φ̃ = arg Φ on Λ (as in the definition of defining sets), then we can estimate the argument of Φ̃Φ in terms of the data (Φ, Λ), so we can apply our results concerning uniqueness sets.

3.10. Defining sets of regular operators. Horváth theorem. We now consider the defining sets problem in some restricted classes of inner functions. We will use the spectral theory language. For r ≥ 1 let Schr(L^{r}, D) denote the class of selfadjoint Schrödinger operators on [0, 1] with an L^{r} potential and Dirichlet boundary condition at 0.

We say that Λ ⊂ ℝ is a defining set for the class Schr(L^{r}, D) if for any two operators in Schr(L^{r}, D) with potentials q and q̃, the equality Θ̃ = Θ on Λ implies q̃ ≡ q, where Θ̃ and Θ are the corresponding Weyl inner functions.

We have a similar definition for the classes Schr(L^{r}, N) of Schrödinger operators with Neumann boundary condition at 0.

Let Θ_{D} denote the standard inner function (1.7), i.e. the Weyl inner function in the case q ≡ 0. From Lemma 3.9 we immediately conclude that

Λ is defining in the class Schr(L^{1}, D) if Λ is a uniqueness set of K^{∞}[Θ_{D}^{2}].

This sufficient condition is not optimal because for regular operators, the function Φ̃ − Φ (see the statement of Lemma 3.9) has some extra smoothness at infinity as follows from the standard asymptotic formulae (see the end of the section), which are getting more precise if we require more regularity of the potential, in particular if we consider the case q ∈ L^{r} with r > 1.

In an interesting paper [22], Horváth gives a complete description of defining sets in terms of uniqueness sets of certain model spaces (or, equivalently, in terms of the completeness problem for exponential functions). The description involves the spaces FL^{r} ≡ FL^{r}(−2, 2), where F stands for the classical Fourier transform (0.1). Recall that

P W_{2} = FL^{2} ⊂ FL^{1} ⊂ Cart_{2} ∩ L^{∞}(ℝ).

Here’s a selection of Horváth’ results. We use the following notation: \sqrt{Λ} = \{z : z^{2} ∈ Λ\}, and \sqrt{Λ} ∪ \{∗, ∗\} means \sqrt{Λ} plus any two points.

(i) Λ is defining in the class Schr(L^{r}, D) iff \sqrt{Λ} ∪ \{∗, ∗\} is a uniqueness set of FL^{r};

(ii) Λ is defining in Schr(L^{r}, N) if \sqrt{Λ} is not a zero set of FL^{r}.

(In the second case, the ”only if” part of Horváth’ theorem comes with some additional condition.)

Let us explain how to prove the ”if” parts of these statements using the methods of this paper. We prove for example (ii).

Proposition. Λ is defining in the class Schr(L^{2}, D) if \sqrt{Λ} ∪ \{∗, ∗\} is a uniqueness set of P W_{2}.

Proof: Let q, q̃ ∈ L^{2}(0, 1). Without loss of generality we will assume that the corresponding Schrödinger operators with boundary conditions (D) at 0 and (N) at 1 are positive. Otherwise, we simply add a large positive constant a to both potentials, and using the transformation

F (z) ↦ F (\sqrt{z^{2} + a^{2}})

for even entire functions we observe that \sqrt{Λ} is a uniqueness set iff \sqrt{Λ + a} is.

Let Θ^{∗} and Θ̃^{∗}(z) be the square root transforms of Θ and Θ̃, the Weyl functions taken with sign minus, see Section 1.8. From the standard asymptotic formula for solutions of a regular Schrödinger equation we obtain

\frac{Θ^{∗}}{S^{2}} = \frac{H̄}{H} on ℝ,  H^{±1} ∈ H^{∞},  (3.14)

and

x[Θ^{∗}(x) − Θ̃^{∗}(x)] ∈ L^{2}(ℝ).  (3.15)

(For convenience we reproduce the standard argument at the end of the proof .)

If Θ̃ = Θ on Λ, then since Θ̃^{∗}(0) = Θ^{∗}(0), we have

Θ^{∗} = Θ̃^{∗} on \{0\} ∪ \sqrt{Λ},

where we regard Θ^{∗} and Θ̃^{∗} as meromorphic functions in the whole plane. By (3.15),

(z − 1)(Θ^{∗} − Θ̃^{∗}) ∈ K[Θ^{∗}Θ̃^{∗}],

so \sqrt{Λ}∪\{0, 1\} is a zero set of some K[Θ^{∗}Θ̃^{∗}] function, and therefore by (3.14) a zero set of some function in K[S^{4}] or P W_{2} . (For zeros in C_{−} we can use the argument with dual reproducing kernels as in Section 3.1.) □

Proof of (3.14)–(3.15). If s > 0, then the solution u_{s}(t) of the IVP

−ü + qu = s^{2}u, u(0) = 0, u̇(0) = 1,

satisfies the integral equation

u_{s}(x) = sin sx + \frac{1}{s} ∫_{0}^{x} cos s(x − t) q(t) u_{s}(t) dt.

Iterating, we have

u_{s}(1) = sin s + \frac{F (s)}{s} + \frac{R(s)}{s^{2}},

where

F (s) = ∫_{0}^{1} cos s(1 − t) sin st q(t) dt,

and

R(s) = ∫_{0}^{1} cos s(1 − x) q(x) dx ∫_{0}^{x} cos s(x − t) q(t) u_{s}(t) dt.

We have an elementary a priori bound

|u_{s}(t)| ≤ C, (s > 0, t ∈ [0, 1]),

so

∀s, |R(s)| ≤ const.

On the other hand, F is basically the Fourier transform of a function on (−1, 1), and

q ∈ L^{2} ⇒ F ∈ L^{2}(ℝ).

We also get the corresponding estimates of u̇_{s}(1). The resulting estimates of Θ imply both statements.

4. Beurling-Malliavin theory

Multiplier theorems

4.1. Tempered inner functions. A meromorphic inner function Θ = e^{iΘ} is called tempered if Θ′ has at most polynomial growth at ±∞:

∃N, Θ′(x) = O (|x|^{N}) , x → ∞.

Theorem. Suppose Θ is a tempered inner function. Then for any meromorphic inner function J and any p > 0,

N^{p}[Θ̄J] ≠ 0  ⇒  ∃n, N^{∞}[b̄^{n}Θ̄J] ≠ 0.

Note that the opposite is trivial: if q > p then N^{q}[Θ̄J] ≠ 0 implies N^{p}[b̄^{n}Θ̄J] ≠ 0 with n = n(p, q). Questions of this type were studied by Dyakonov [13] who was the first to observe the analogy with the Beurling-Malliavin multiplier theorem.

The proof for p = 2 is elementary: if F ∈ N[Θ̄J], then JF ∈ K_{Θ} and by (1.6) and (1.4),

∑_{λ∈σ(Θ)} \frac{|F (λ)|^{2}}{|Θ′(λ)|} ≍ ∥F ∥^{2}.

Thus |F (λ)| ≲ |Θ′(λ)|, and this is true for all λ ∈ ℝ because we can replace Θ with e^{−iα}Θ. It follows that (z + i)^{−n}F (z) ∈ N^{∞}[b̄^{n}Θ̄J].

We will derive the theorem from the following special case of Carleson’s type embedding theorem of Treil and Volberg [40]. For a given meromorphic inner function Θ denote

d(x) = dist{x, {|Θ| = 0.5}), (x ∈ ℝ).

Claim: the measure ν_{x} = d(x)δ_{x} is a Carleson measure for K_{Θ}^{p}, i.e.

K_{Θ}^{p} ⊂ L^{p}(ν),

where the norm of the embedding depends only on p. In other word, if F ∈ K_{Θ}^{p}, then

d(x) |F (x)|^{p} ≤ const,  (x ∈ ℝ).  (4.1)

Lemma. If Θ is tempered, then there is an N such that

dist(x, {|Φ| = 0.5}) ≳ (1 + |x|)^{−N}.

Proof: We can only consider the case of Blaschke products. If

Θ = B_{Λ} = ∏ ϵ_{λ} \frac{z − λ}{z − λ̄},

then we have

Θ′(x) = ∑ \frac{2ℑλ}{|x − λ|^{2}}  (4.2)

and

− log |Θ(z)| ≍ ∑ \frac{ℑλ · ℑz}{|z − λ̄|^{2}} .  (4.3)

Indeed,

− log |Θ(z)|^{2} = ∑ log |\frac{z − λ̄}{z − λ}|^{2}

≍ ∑ [1 − \frac{|z − λ|^{2}}{|z − λ̄|^{2}} ] (see below)

= ∑ \frac{(z − z̄)(λ − λ̄)}{|z − λ̄|^{2}}

We want to show that if z = x + iy and y ≪ |x|^{−N}, then (4.3)< const. For each x consider the Stolz sector of some fixed angle at x of height x^{−N} and observe that this sector does not contain any λ, for otherwise, the corresponding term in (4.2) would be of the order 1/ℑλ ≥ x^{N}. This justifies the ”see below” item, and makes the estimate of (4.3) in terms of (4.2) obvious. □

Proof of Theorem. Suppose N^{p}[Θ̄J] ≠ 0, so

Θ̄JF = Ḡ, F, G ∈ H^{p} ∩ C^{ω}(ℝ).

We have

JF ∈ K_{Θ}^{p}

so by the lemma and by (4.1) we have

|F (x)| ≲ 1 + |x|^{N}

for all x ∈ ℝ and some N. It follows that (z + i)^{−N}F (z) ∈ N^{∞}[b̄^{N} Θ̄J]. □

Corollary. Let U = Θ̄J = e^{iγ} and let Θ be tempered. If γ is the sum of a bounded and a decreasing functions, then N^{∞}[b̄^{n}U] ≠ 0 for some n.

Note that in the case of a general bounded γ, we can not multiply down to H^{∞} elements of N[U] even by using factors like S̄.

Also note that in the statement of the theorem one can give explicit bounds on n in terms of the growth of |Θ′(x)|. For example, if U = Θ̄J and Θ′ ≤ 1, then N[U] ⊂ N^{∞}[U].

4.2. Beurling-Malliavin multiplier theorem.

Theorem. Suppose Θ is a meromorphic inner function satisfying |Θ′| ≤ const. Then for any meromorphic inner function J, we have

N^{+}[Θ̄J] ≠ 0  ⇒  ∀ϵ, N^{∞}[S̄^{ϵ}Θ̄J] ≠ 0.

This theorem follows from the Beurling-Malliavin multiplier theorem [4]: if W is an outer function, then

z^{−1} log W (z) ∈ D(C_{+})  ⇒  W ∈ (BM).  (4.4)

Here D(C_{+}) is the notation for the usual Dirichlet space in the halfplane, and by definition, W is a Beurling-Malliavin multiplier, or W ∈ (BM), if

∀ϵ > 0, ∃G ∈ K[S^{ϵ}], W G ∈ L^{2}(ℝ).

Note that if |W | ≤ |W_{1}| and W_{1} ∈ (BM), then W ∈ (BM).

Lemma. If W ∈ C^{ω}(ℝ), |W | ≥ 1, and (arg W)′ ≤ const on ℝ, then W ∈ (BM).

Proof: We will use some ideas from the proof of Theorem 64 in [12]. We can assume |W (0)| = 1. Otherwise, multiply W by (z + i) to get W (∞) = ∞. In this case there’s a global minimum, which we can take for 0. Denote

z^{−1} log W (z) = u(z) + iv(z),

where u and v are real-valued functions. Then we have

x^{−1}u(x) ∈ L^{1}(ℝ),  x^{−1}u(x) ≥ 0 on ℝ.  (4.5)

Since arg W ∈ L̃_{Π}^{1} and (arg W)′ ≤ const on ℝ, by Lemma 3.4 we have

arg W (x) = o(|x|),  x ∈ ℝ, x → ∞,  (4.6)

and it follows that v is a bounded function in C_{+}. For r > 0 let D(r) denote the semidics {|z| < r} ∩ C_{+}. We have

∥u + iv∥_{D}^{2} = lim_{r→∞} ∫_{∂D(r)} udv,

and

∫_{∂D(r)} udv = ∫_{−r}^{r} uv′dx − uv|_{−r}^{r} + rI′(r), I(r) := \frac{1}{2} ∫_{0}^{π} v^{2} (re^{iθ}) dθ.

By (4.5) and (4.6), the integrals ∫ uv′dx are uniformly bounded from above:

∫_{−r}^{r} uv′dx = ∫_{−r}^{r} \frac{u(x)}{x}(arg W)′(x)dx − ∫_{−r}^{r} \frac{u(x)}{x} · \frac{arg W (x)}{x}dx < const.

It remains to show that

lim inf_{r→∞} A(r) < ∞, A(r) := rI′(r) − uv|_{−r}^{r}.

Suppose A(r) ≥ 1 for all r ≫ 1. Then since v is bounded, we have

I′(r) ≥ \frac{1}{r} − const\frac{u(r) + u(−r)}{r}.

By (4.5), this contradicts the uniform boundedness of I(r). □

Corollary. If W ∈ K_{Θ}^{+} and (arg Θ)′ ≤ const, then W ∈ (BM).

Proof: We have W Θ̄ = H̄ for some H ∈ N^{+}. Define

W_{1} = W H + Θ.

Clearly, W_{1} ∈ N^{+}, and since

Θ̄^{2}W_{1} = Θ̄W Θ̄H + Θ̄ = H̄W̄ + Θ̄ = W̄_{1},

we have W_{1} ∈ K^{+}[Θ^{2}]. Notice that

|W_{1}| = |W W̄ Θ + Θ| = 1 + |W |^{2} ≥ 1.

Also, |W | ≤ |W_{1}|. Since (arg Θ)′ ≤ const, from the equation Θ̄^{2}W_{1} = W̄_{1}Φ̄, where Φ is an inner function, we obtain (arg W_{1})′ ≤ const. By the previous lemma, W_{1} ∈ (BM), and therefore W ∈ (BM). □

Proof of Theorem. Take an outer function W ∈ N^{+}[Θ̄J]. Then W ∈ K_{Θ}^{+} and by the last corollary, W ∈ (BM), and therefore W G ∈ H^{2} for some G ∈ N^{+}[S̄^{ϵ}]. It then follows that

W G ∈ N^{+}[S̄^{ϵ}Θ̄J] ∩ H^{2} = N[S̄^{ϵ}Θ̄J].

It remains to multiply down to H^{∞}, which is possible by Theorem 4.1. □

N^{+}-part of the Beurling-Malliavin theory

In this part of the section we will assume

γ′ ≥ −const.  (4.7)

We’ll give a metric criterion for (non)triviality of the Toeplitz kernel N^{+}[e^{iγ}] up to a gap S^{±ϵ}. Recall the basic criterion: N^{+}[e^{iγ}] ≠ 0 iff

γ = −α + h̃  (4.8)

for some increasing function α ∈ C^{ω}(ℝ) and some h ∈ L_{Π}^{1}. As usual we denote U = e^{iγ}.

Lemma. Suppose γ′ ≥ −const. Then ∀ϵ > 0, N^{+}[US^{ϵ}] = 0 unless γ(∓∞) = ±∞.

Proof: If N^{+}[US^{ϵ}] ≠ 0, then by (4.8) γ + ϵx + α = h̃, so (h̃)′ ≥ −const and of course h̃ ∈ L_{Π}^{o(1,∞)}. By Lemma 3.4 it then follows that h̃(x) = o(x) as x → ±∞, so γ + ϵx + α = o(x), which is possible only if γ(∓∞) = ±∞. □

4.3. Beurling-Malliavin intervals. Suppose a continuous function γ : ℝ → ℝ satisfies

γ(−∞) = +∞,  γ(+∞) = −∞.  (4.9)

The family BM(γ) is defined as the collection of the components of the open set {γ^{⋆} ≠ γ}, where

γ^{⋆}(x) = max_{[x,+∞]} γ.

For an interval l = [a, b] ⊂ ℝ_{+} or ⊂ ℝ_{−} we write |l| for the Euclidian length, and δ(l) for the distance from the origin. A family of finite disjoint intervals {l} is called long if

∑_{δ(l)≥1} \frac{|l|^{2}}{δ(l)^{2}} = ∞.

Otherwise, we call the family short.

Theorem. Suppose γ′ > −const.

(i) If γ ∉ (4.9), or if γ ∈ (4.9) but the family BM(γ) is long, then

∀ϵ > 0, N^{+}[S^{ϵ}U] = 0.

(ii) If γ ∈ (4.9) and BM(γ) is short, then

∀ϵ > 0, N^{+}[S̄^{ϵ}U] ≠ 0.

The first statement corresponds to the ”second Beurling-Malliavin theorem” [5], and statement (ii) to the so called ”little multiplier theorem”, see [25], [19].

4.4. Second Beurling-Malliavin theorem. The first part of Theorem 4.3 follows from a more general fact; we don’t need to assume (4.7). For an interval l ⊂ ℝ we denote

Δ_{l}^{∗}[γ] = inf_{l′′} γ − sup_{l′} γ,

where l′ and l′′ are the left and the right adjacent intervals of length |l|.

Theorem. If ∃c > 0 and a long family of intervals {l} such that

Δ_{l}^{∗}[γ] ≥ c|l|,  (4.10)

then N^{+}[U] = 0.

A simple standard argument shows that we can assume without loss of generality that all intervals satisfy the inequality 10|l| < δ(l), and that the multiplicity of the covering {10l} is finite. (The interval 10l is concentric with l and has length 10|l|.)

The idea of the proof is quite simple. According to the basic criterion (4.8) we have to exclude the possibility

γ + α = h̃, α ↑, h ∈ L_{Π}^{1}.

Since α is increasing we have

Δ_{l}^{∗}[h̃] ≳ |l|.

Suppose we can localize this estimate to each function h̃_{l}, where h_{l} is the restriction of h to the interval 10l. Choosing A ≍ |l| and applying the weak type inequality, we have

\frac{|l|}{δ(l)^{2}} ≲ Π\{|h̃_{l}| > A\} ≲ \frac{∥h_{l}∥_{Π}}{A} ,  (4.11)

so

\frac{|l|^{2}}{δ(l)^{2}} ≲ ∥h_{l}∥_{Π}.  (4.12)

Summing up over l’s we arrive to a contradiction:

∞ = ∑ \frac{|l|^{2}}{δ(l)^{2}} ≲ ∑ ∥h_{l}∥_{Π} ≲ ∥h∥_{Π} < ∞.

Proof of Theorem. For an interval l we denote by Q_{l} its Carleson square:

Q_{l} = {z : x ∈ l, |l| < y < 2|l|}.

Also, denote

H(z) = ∫_{ℝ} \frac{h^{−}(t)dt}{(t − z)^{2}} , (z ∈ C_{+}),

where as usual h_{−} = max{0, −h}. We claim that if the estimate (4.12) does not hold for some interval l in our family, then

|H| ≳ 1  on Q_{l}.  (4.13)

To see this we observe that the argument (4.11) is valid unless there is an interval l_{1}, l ⊂ l_{1} ⊂ l ∪ l′ ∪ l′′, such that

Δ_{l_{1}} f̃ ≥ (c/2)|l|, f := h − h_{l}.

Let’s assume (for simplicity of notation) l_{1} = l, so Δf̃ ≡ Δ_{l}f̃ ≳ l. Represent f = f^{+} − f^{−}, f^{±} ≥ 0. The functions f̃^{±} are decreasing on l:

f̃′_{±}(x) = −\frac{1}{π} ∫_{ℝ\(10l)} \frac{f^{±}(t)dt}{(t − x)^{2}} > 0, (x ∈ l).

It follows that −Δf̃^{−} ≳ |l|, and so there is a point x_{∗} ∈ l such that

\frac{1}{π} ∫ \frac{f^{−}(t)dt}{(t − x_{∗})^{2}} = −f̃′_{−}(x_{∗}) = −\frac{Δf̃^{−}}{|l|} ≳ 1.

If z ∈ Q_{l}, then

|∫ \frac{f^{−}(t)dt}{(t − z)^{2}} | ≥ ∫ ℜ [\frac{1}{(t − z)^{2}} ] f^{−}(t)dt ≍ ∫ \frac{f^{−}(t)dt}{(t − x_{∗})^{2}} ≳ 1.

On the other hand, if z ∈ Q_{l} and if (4.12) is not true, then

|∫ \frac{h_{l}^{−}(t)dt}{(t − z)^{2}} | ≲ \frac{1}{|l|^{2}} ∫_{(10l)} |h| ≍ \frac{δ(l)^{2}}{|l|^{2}} ∫_{(10l)} |h|dΠ ≪ \frac{δ(l)^{2}}{|l|^{2}} · \frac{|l|^{2}}{δ(l)^{2}} = 1,

and we get

|H(z)| = |∫ \frac{f^{−}(t) + h_{l}^{−}(t)}{(t − z)^{2}}dt| ≥ |∫ \frac{f^{−}(t)dt}{(t − z)^{2}} | − |∫ \frac{h_{l}^{−}(t)dt}{(t − z)^{2}} | ≳ 1.

To finish the proof of the theorem it remains to show that

∑_{l∈(∗)} \frac{|l|^{2}}{δ(l)^{2}} < ∞,

where we write l ∈ (∗) if (4.13) holds for l. Denote ψ = ∑_{l∈(∗)} |l|χ_{l}, so

∑_{l∈(∗)} \frac{|l|^{2}}{δ(l)^{2}} ≍ ∫ \frac{ψ(t)dt}{1 + t^{2}}.

Then we have

∫ \frac{ψ(t)}{1 + t^{2}} dt ≲ ∫_{1}^{∞} \frac{dA}{A^{3}} ∫_{−A}^{A} ψ(t)dt.

Fix C ≫ 1 such that

∫_{|t|≥C} \frac{h^{−}(t)dt}{1 + t^{2}} ≪ 1.

We claim that

∫_{−A}^{A} ψ(t)dt ≲ ∫_{−CA}^{CA} h^{−}(t)dt,

so we have

∫ \frac{ψ(t)}{1 + t^{2}} dt ≲ ∫_{1}^{∞} \frac{dA}{A^{3}} ∫_{−CA}^{CA} h^{−}(t)dt ≲ ∥h∥_{Π}

(and we are done). To prove the claim, fix A and consider the 2D Hilbert transform

H_{A}(z) = ∫_{−CA}^{CA} \frac{h^{−}(t)dt}{(t − z)^{2}} .

If l ⊂ (−A, A) and l ∈ (∗), then

|H_{A}(z)| ≍ |H(z)| ≳ 1, z ∈ Q_{l},

by the choice of C. Applying the weak-L^{1} estimate for the Hilbert transform, we get

∫_{−A}^{A} ψ(t)dt ≲ Area(|H_{A}| ≳ 1) ≲ ∫_{−CA}^{CA} h^{−}(t)dt. □

4.5. Little multiplier theorem. We now turn to the proof of the second part of Theorem 4.3. We assume (4.7) and γ(∓∞) = ±∞. The function γ^{⋆} and the family BM(γ) were defined in section 4.3. Note that γ^{⋆} is decreasing and γ^{⋆} − γ ≥ 0.

Lemma. If the family BM(γ) is short, then γ^{⋆} − γ ∈ L_{Π}^{1}.

Proof:

∫_{l} \frac{γ^{⋆} − γ}{1 + x^{2}} dx ≲ \frac{1}{δ(l)^{2}} ∫_{0}^{|l|} tdt ≍ \frac{|l|^{2}}{δ(l)^{2}} .

□

The proof of the little multiplier is extremely simple if we settle for a slightly weaker statement:

∑_{BM(γ)} \frac{|l|^{2} log^{+} |l|}{δ(l)^{2}} < ∞  ⇒  N^{+}[U] ≠ 0.

Indeed, the last computation shows that in this case we have

∫ (γ^{⋆} − γ) log^{+}(γ^{⋆} − γ) dΠ < ∞,

which is a (necessary and) sufficient condition for (γ^{⋆} − γ)˜ ∈ L_{Π}^{1}, see [42]. We have a representation

γ = γ^{⋆} + (γ − γ^{⋆}),

where the first term is decreasing and the second one is in L̃_{Π}^{1}, so N^{+}[U] ≠ 0 by the basic criterion.

Main Lemma. If the family BM(γ) is short, then for any given ϵ > 0 there is a function β such that β′ ≤ ϵ near ±∞ and

γ^{⋆} − γ + β ∈ L̃_{Π}^{1}.

The little multiplier theorem now follows immediately:

γ(x) − ϵx = (γ − γ^{⋆} − β) + (β − ϵx) + γ^{⋆}.

The first term is in L̃_{Π}^{1}, and the last two terms are decreasing at infinities.

Proof of Main Lemma. Denote f = γ^{⋆} − γ, so f is a non-negative function in L_{Π}^{1}, f′ ≤ const, and the family L = BM(γ) of the components of {f ≠ 0} satisfies

∑_{l∈L} \frac{|l|^{2}}{δ(l)^{2}} < ∞.

These are the only properties of f that will be used. We will also need the following notation: for an interval l = [a, b] we define its ”tent” function

T_{l}(x) = \begin{cases}x − a,  a ≤ x ≤ (a + b)/2, \\ b − x,  (a + b)/2 ≤ x ≤ b, \\ 0,  x ∈ ℝ \ l. \end{cases}

Note that

∥T_{l}∥_{Π} ≳ \frac{|l|^{2}}{(δ(l) + |l|)^{2}} .  (4.14)

We will construct disjoint intervals l_{n} such that {f ≠ 0} ⊂ ∪ l_{n},

∑_{n} \frac{|l_{n}|^{2}}{δ(l_{n})^{2}} < ∞,  (4.15)

and

∀n ∃ϵ_{n} ∈ [0, ϵ],  ∫_{l_{n}}(f − ϵ_{n}T_{l_{n}}) dΠ = 0.  (4.16)

Let us show that the existence of such intervals l_{n} implies the main lemma. We will use the easier part of the atomic decomposition technique of Hardy spaces, see [11].

We define

β = − ∑_{n} ϵ_{n}T_{l_{n}}, g = f + β,

so β is in Lip(ϵ), and all we need to check is that g̃ ∈ L_{Π}^{1} or, in other words, that g belongs to the real Hardy space H_{Π}^{1}(ℝ). We can represent g as follows:

g = ∑ χ_{n}g = ∑ λ_{n} \frac{χ_{n}g}{λ_{n}} := ∑ λ_{n}A_{n},

where χ_{n} is the characteristic function of l_{n}, and we choose

λ_{n} = Π(l_{n}) ∥χ_{n}g∥_{∞}.

It is clear that the functions A_{n} = λ_{n}^{−1}(χ_{n}g) are ”atoms”:

∫ A_{n}dΠ = \frac{1}{λ_{n}} ∫_{l_{n}} gdΠ = 0 by (4.16),

and

∥A_{n}∥_{∞} = \frac{∥χ_{n}g∥_{∞}}{λ_{n}} = \frac{1}{Π(l_{n})}.

By (4.15) we also have

∑ λ_{n} ≲ ∑ Π(l_{n})|l_{n}| ≍ ∑ \frac{|l_{n}|^{2}}{δ(l_{n})^{2}} < ∞.

It follows that ∑ λ_{n}A_{n} ∈ H_{Π}^{1}(ℝ), see [11].

To finish the proof it remains to describe the construction of the intervals l̃_{n} and the slopes ϵ_{n}. We consider the case where all intervals l ∈ L are in (1, +∞). The construction for intervals in [−∞, −1) is similar.

Suppose the left endpoint a_{n} = a of l_{n} has been constructed, and a is also the left endpoint of some interval l = (a, b(l)) ∈ L. (To start the induction we take the leftmost endpoint for a_{1}.) Consider the function

F (b) ≡ F_{ϵ}(b) = ∫_{a}^{b} [f − ϵT_{(a,b)}] dΠ,

and define

b_{n} = min{b ≥ b(l) : f(b) = 0, F (b) ≤ 0}.

For example, if we already have F (b(l)) ≤ 0, then b_{n} = b(l). Since f ∈ L_{Π}^{1}, we have F (+∞) = −∞ and so b_{n} < ∞. We also define ϵ_{n} from the equation

∫_{a_{n}}^{b_{n}} [f − ϵ_{n}T_{(a_{n},b_{n})}] dΠ = 0.

Finally, we define a_{n+1} as the leftmost endpoint of all intervals l ∈ L to the right of l_{n}.

It is clear from the construction that the intervals l_{n} cover {f ≠ 0}, that all ϵ_{n}’s are ≤ ϵ, and that we have (4.16). Let us check (4.15). We consider three types of intervals l_{n}:

(a) F (b_{n}) < 0 but ∃l ∈ L such that l ⊂ l_{n} and |l|/δ(l) ≍ |l_{n}|/δ(l_{n}),

(b) F (b_{n}) = 0,

(c) other intervals.

Property (4.15) is obvious for the group (a). For the group (b), we use (4.14):

∑_{(b)} \frac{|l_{n}|^{2}}{(δ(l_{n}) + |l_{n}|)^{2}} ≲ \frac{1}{ϵ} ∫_{∪_{(a)}l_{n}} fdΠ < ∞.

The argument for the group (c) is similar as long as we can show that the slopes are > ϵ/2, i.e. F_{ϵ/2}(b_{n}) > 0. Since F (b_{n}) < 0, b_{n} is by construction the right endpoint of some interval l ∈ L, and since l_{n} ∉ (a) we have |l| ≪ |l_{n}| . Let c be the left endpoint of the above l; by construction, F (c) > 0. We have

F_{ϵ/2}(b_{n}) = (∫_{a_{n}}^{c} + ∫_{c}^{b_{n}}) [f − \frac{ϵ}{2}T_{(a,b)}] dΠ > ∫_{a_{n}}^{c} [f − ϵT_{(a,c)}] dΠ = F (c) > 0.

Beurling-Malliavin density

4.6. Radius of completeness. Let Λ ⊂ ℝ and let E_{Λ} denote the family of exponential functions {e^{iλt} : λ ∈ Λ}. By definition, the radius of completeness of the family E_{Λ} is the number

R(Λ) = sup {a : E_{Λ} is complete in L^{2}(0, a)} .

In terms of Toeplitz kernels, by Proposition 3.2 we have

R(Λ) = sup {a : N[S̄^{a}J_{Λ}] = 0} ,

where J_{Λ} denotes some/any meromorphic inner function J such that {J = 1} = Λ. By the Beurling-Malliavin multiplier theorem we also have

R(Λ) = sup {a : N^{+}[S̄^{a}J_{Λ}] = 0} .

The combination of the second Beurling-Malliavin and the little multiplier theorems, see Theorem 4.3, then gives the following metric characterization of R(Λ). By definition, the Beurling-Malliavin density of Λ is the number

d_{BM}(Λ) = inf{a : γ_{a}(±∞) = ∓∞ and BM[γ_{a}] is short}, γ_{a} := 2πn_{Λ} − at.

Corollary. R(Λ) = d_{BM}(Λ).

4.7. General transition parameter problems. Let M be a unimodular function with non-decreasing continuous argument; we call it a gap function. For a given unimodular function U = e^{iγ} we want to compute the critical exponent

sup {a : N[M̄^{a}U] = 0} .

This number of course can be ±∞. The Beurling-Malliavin density theorem corresponds to the case M = S and U = J (or S̄^{c}J). More generally, Theorems 4.2 and 4.3 allow us to compute the transition parameter in the case M = S and U = Θ̄J where |Θ′| ≤ const.

Of course, one can state similar problems concerning families of Toeplitz kernels in other functional spaces. Theorem 4.1 states that the critical exponent is the same in all H^{p}-spaces if both M and Θ are tempered and the ”gap” is wide enough: N^{∞}[b^{N} M̄^{ϵ}] ≠ 0 for all ϵ > 0 and ∀N.

One could try to generalize the Beurling-Malliavin theory (M = S) to arbitrary gap functions. As a first step it is natural to consider the ”standard” gap functions S^{(α)} and S_{+}^{(α)}, α > 0, defined as follows:

S^{(α)}(x) = \begin{cases}S(x^{α}),  x > 0, \\ S(−|x|^{α}), x < 0, \end{cases}  S_{+}^{(α)}(x) = \begin{cases}S(x^{α}), x > 0, \\ S(0), x < 0. \end{cases}

Note that these functions satisfy identities like

S^{(α)}(kx) = [S^{(α)}(x)]^{k^{α}} , (k > 0).

For each standard gap function, one would expect to have some kind of a Beurling-Malliavin theory, i.e. a combination of theorems that allow to express the transition parameter in terms of the Beurling-Malliavin intervals under an appropriate growth restriction on γ.

Here is a typical example of a transition parameter problem with a standard gap function.

Example 1. Let Ai(x) denote the usual Airy function: it is a solution of the equation y′′ = xy, which is L^{2} at +∞. It is well-known that the Airy function is entire and also

Ai(x) = \sqrt{x}F (\frac{2}{3}ix^{3/2})

for some solution F of the Bessel equation of order 1/3. If x = −λ and λ > 0, then we have

Ai(−λ) = i\sqrt{λ}F (\frac{2}{3}λ^{3/2}) ∼ λ^{−1/4} cos (\frac{2}{3}λ^{3/2} + const) , λ → +∞.

(4.17)

Given Λ ⊂ ℝ we ask if the family of shifts

E_{Λ} = {Ai(t − λ) : λ ∈ Λ}

is complete in L^{2}(ℝ_{+}). Note that u_{λ}(t) = Ai(t − λ) is an L^{2}-solution of the Schrödinger equation

−ü + tu = λu, t ∈ ℝ_{+},

so we can apply our general approach, see Section 3.1.

Claim. Up to a finite dimensional gap, the family E_{Λ} is complete in L^{2}(ℝ_{+}) iff

N[M̄J_{Λ}] = 0, M = [S_{+}^{(3/2)}]^{2/3} .

Proof: According to Section 3.1 and Theorem 4.1, the criterion for completeness (up to a finite dimensional gap) is N[Θ̄J_{Λ}] = 0, where Θ is the Weyl inner function. We can replace Θ by M because of the asymptotic formula (4.17) for u_{λ}(0) = Ai(−λ) and a similar formula for u̇_{λ}(0). □

Remark. This completeness problem, which involves shifts of a given function in L^{2}(ℝ_{+})) is different from the well-known Wiener problem concerning shifts in L^{2}(ℝ). The latter is essentially the problem concerning exponential families in weighted L^{2}-spaces; it can also be restated in terms of Toeplitz kernels.

Let us now state the transition parameter problem. Given Λ, denote by E_{Λ}^{(a)} the family {u_{λ} : λ ∈ Λ} of L^{2}-solutions of the Schrödinger equation with potential q(t) = t/a, (a > 0). We want to compute the ”radius of completeness” R(Λ), i.e. the critical value of a such that E_{Λ}^{(a)} is complete in L^{2}(ℝ_{+}). In terms of Toeplitz kernels we have

R(Λ) = sup {a : N[M̄^{a}J_{Λ}] = 0} , M = [S_{+}^{(3/2)}]^{2/3} .

This is similar to the Beurling-Malliavin situation, which can be reformulated as the completeness problem for the solutions of the Schrödinger equation with q ≡ 0 on [0, a] :

R(Λ) = sup {a : N[M̄^{a}J_{Λ}] = 0} , M = S_{+}^{(1/2)}.

In the Airy situation, the parameter a characterizes the ”size” of the singularity, which plays the same role as the length of the interval in the regular case.

Transition parameter problems arise also in connection with the spectral theory problems that we discussed in Section 3.

Example 2. Let L is a regular Schrödinger operator on [0,1] and let Λ ⊂ σ(L). We want to characterize the numbers c such that

(i) the potential on [0, c] and the partial spectrum Λ determine L, (ii) the potential on [0, c] and the spectral measure on Λ determine L, see Section 3.8. According to the results of Section 3.9 (and Theorem 4.1), these questions are equivalent to the transition parameter problems with M = S_{+}^{(1/2)} – to find the values of

inf {c : N [M̄^{2(1−c)}J_{Λ}] = 0} , inf {c : N [M̄^{2(1−c)}J_{Λ}^{2}] = 0}

in the cases (i) and (ii) respectively. Indeed, in the case (i), Λ has to be defining for Φ, the Weyl function corresponding to the restriction of the potential to [c, 1], which means that N[Φ̄^{2}J_{Λ}] = 0 within the admissible gap. Also, we can replace Φ by M^{(1−c)}. In the case (ii), we use the same argument for the divisor 2χ_{Λ}.

4.8. Square root transformation. The following construction and its corollaries in this and the next subsections are meant to give some isea of what the Beurling-Malliavin theories of standard gap functions may look like.

Let U = e^{iγ} be a unimodular function such that γ = 0 on ℝ_{−}, and let U_{∗} = e^{iγ_{∗}} be a unimodular function with an odd argument related to γ by the equation

γ_{∗}(t) = γ(t^{2}), (t ≥ 0).

Proposition.

N^{∞}[U] ≠ 0 ⇔ N^{∞}[U_{∗}] ≠ 0.

Proof: ⇐ Suppose we have U_{∗}H = Ḡ on ℝ for some H, G ∈ H^{∞}. Then we also have U_{∗}H^{♭} = Ḡ^{♭}, where we use the notation

H^{♭}(z) = \overline{H(−z̄)}.

(Note U_{∗} = U_{∗}^{♭}.) Thus we have

U_{∗}F = F̄^{♭}, F := H + G^{♭}.

Consider now the functions f, g ∈ H^{∞}(C_{+}),

f(z) = F (\sqrt{z}), g(z) = F̄(\sqrt{z̄}),

where the square root denotes the conformal map C \ ℝ_{+} → C_{+}. Let us check that Uf = ḡ on ℝ. If t > 0, then f(−t^{2}) = F (it) and g(−t^{2}) = F̄(it), so

\frac{ḡ(−t^{2})}{f(−t^{2})} = 1 = U(−t^{2}).

On the other hand, f(t^{2}) = F (t) and g(t^{2}) = F̄(−t) = F^{♭}(t), and therefore

\frac{ḡ(t^{2})}{f(t^{2})} = \frac{F̄^{♭}(t)}{F (t)} = U_{∗}(t) = U(t^{2}).

⇒ Suppose Uf = ḡ on ℝ for some f, g ∈ H^{∞}. Since U ≡ 1 on ℝ_{−}, the analytic functions f ∈ H^{∞}(C_{+}) and g^{#} ∈ H^{∞}(C_{−}) match on ℝ_{+} and therefore define a function in C\ℝ_{+}. Applying the square root transformation, we get F ∈ H^{∞}(C_{+}),

F (z) = \begin{cases}f(z^{2}), ℜ z > 0 \\ g^{#}(z^{2}), ℜ z < 0 \end{cases}

Let us check that U_{∗}F = F̄^{♭} on ℝ. If x > 0, then F (x) = f(x^{2}), F̄^{♭}(x) = F (−x) = ḡ(x^{2}), and

\frac{F̄^{♭}(x)}{F (x)} = \frac{ḡ(x^{2})}{f(x^{2})} = U(x^{2}) = U_{∗}(x).

On the other hand, if x < 0, then F (x) = ḡ(x^{2}), F̄^{♭}(x) = F (−x) = f(x^{2}), and

\frac{F̄^{♭}(x)}{F (x)} = \frac{f(x^{2})}{ḡ(x^{2})} = \frac{1}{U(x^{2})} = U_{∗}(x).

□

The square root transformation makes it possible to derive the Beurling-Malliavin theory of the gap function M = S_{+}^{(1/2)} from Theorems 4.2-4.3 (we also use Theorem 4.1).

Corollary. Let U = e^{iγ} = Θ̄J and suppose that both Θ and J have bounded arguments at −∞. Suppose in addition

(arg Θ)′(t) ≤ \frac{const}{\sqrt{t}}, t → +∞.

(i) If γ(t) ↛ −∞ as t → +∞, or if γ(+∞) = −∞ but the family BM(γ) is long, then

∀ϵ > 0, N [(S_{+}^{(1/2)})^{ϵ} U] = 0.

(ii) If γ(+∞) = −∞ and the family BM(γ) is short, then

∀ϵ > 0, N [\binom{}{S̄_{+}^{(1/2)}}^{ϵ} U] ≠ 0.

Here we consider only the Beurling-Malliavin intervals in ℝ_{+}, and the meaning of the terms ”long” and ”short” is the same as in Section 4.3.

Example. Let L be the Schrödinger operator on ℝ with potential q(t) = t^{2}/4 (”quantum harmonic oscillator”) and let Λ ⊂ σ(L) = N − \frac{1}{2}. We want to find the critical value c_{∗} of real numbers c such that q on (−∞, −c) and the spectral measure on Λ (including the numbering of eigenvalues) determine L, see Section 3.8. We can explicitely compute c_{∗} in terms of Beurling-Malliavin intervals if the set Λ satisfies the inequality

#(Λ ∩ l) ≥ \frac{|l|}{2} − const  (4.18)

for all intervals l = [a, b] ⊂ ℝ_{+} such that b ≤ a + \sqrt{a}. Claim:

c_{∗} = inf\{c : γ_{c}(+∞) = −∞ and BM[γ_{c}] is short\}, γ_{c} := 2n_{Λ}(t)−t−2c\sqrt{t}.

Sketch of Proof: Let Θ be the Weyl inner function corresponding to the restriction of the potential to ℝ_{+}, so Θ^{2} is essentially the same (i.e. up to a finite dimensional gap) as S_{+}. Then the Weyl inner function corresponding to (−c, ∞) is essentially the same as Φ = ΘM^{c}, where M = S_{+}^{(1/2)}. The data determine L if the divisor 2χ_{Λ} is defining for K[Φ^{2}], i.e. (again up to a finite dimensional gap)

N[M̄^{2c}U] = 0, U = S̄_{+}J_{Λ}^{2}.

Note that

γ(t) ≡ arg U(t) = 2n_{Λ}(t) − t + O(1), (t > 0).

To apply the corollary, we need to make sure that γ(t) does not drop by more than a constant on each interval l described in the statement. This is exactly our condition (4.18). □

Let us state another corollary, which gives a necessary condition for the nontriviality of a Toeplitz kernel in the ”one-sided” situation.

Corollary. Let U = e^{iγ} and γ = 0 on ℝ_{−}. If N^{∞}[U] ≠ 0, then

∫_{\{γ>A\}} \frac{dt}{t\sqrt{t}} = \frac{o(1)}{A},  A → +∞,  (4.19)

in particular

γ^{+} ∈ L^{p} (\frac{dt}{1 + t\sqrt{t}}) , (0 < p < 1).

Proof: Consider the square root transform U_{∗}. By the basic criterion, N^{+}[U_{∗}] ≠ 0 implies

Π\{γ_{∗} > A\} = \frac{o(1)}{A}, A → +∞.

□

One can state similar results for tempered γ’s satisfying γ = O(1) at −∞: if N^{p}[U] ≠ 0, then we have (4.19) at +∞.

4.9. Final thoughts. Except for the ”two-sided” case M = S and the ”one-sided” case M = S_{+}^{(1/2)}, the complete Beurling-Malliavin theory of the standard gap functions is not known. Here we mention some preliminary considerations.

Two facts seem to be certain. First, the theories should be different in the subexponential case (α < 1 for two-sided gap functions and α < 1/2 for one-sided functions) and superexponential case (α > 1 and α > 1/2 respectively). Second, the role of the Smirnov class N^{+} in the case S^{(α)}, α ≠ 1, is not the same as in the classical case M = S. One should probably introduce appropriate ”Smirnov classes” for all α < 1, e.g. the preimage of N^{+} under the square root transformation in the case α = 1/2, cf. the last corollary.

Subexponential case. We have the following partial results. From the last corollary we immediately derive

Corollary. Let M = S^{(1/2)} and U = e^{iγ}. Suppose

γ′(x) ≥ −\frac{C}{\sqrt{|x|}}, x → ±∞.

If γ(∓∞) = ±∞ and the family BM(γ) is short, then

∀ϵ > 0, N^{∞} [M̄^{ϵ}U] ≠ 0.

Applying the square root transformation one more time we get

Corollary. Let M = S^{(1/4)} and U = e^{iγ}. Suppose

γ′(x) ≥ −C|x|^{−3/4}, x → ±∞.

If γ(∓∞) = ±∞ and the family BM(γ) is short, then

∀ϵ > 0, N^{∞} [M̄^{ϵ}U] ≠ 0.

These facts should extend to all S^{(α)} with α < 1. The ”converse” statements (if true) should follow from Theorem 4.4 in appropriate ”Smirnov classes”.

Superexponential case. We only discuss the gap function S_{+}, which corresponds to the two-sided case M = S^{(2)}. The proof of Thorem 4.3 applies verbatim to give the following criterion in the class N^{+}.

Proposition. Let U = e^{iγ}. Suppose γ is bounded on ℝ_{−} and γ′ ≥ −const on ℝ_{+}.

(i) If γ ↛ −∞ at +∞, or if γ(+∞) = −∞ but BM(γ) is long, then

∀ϵ > 0, N^{+} [S_{+}^{ϵ}U] = 0.

(ii) If γ(+∞) = −∞ and BM(γ) is short, then

∀ϵ > 0, N^{+} [S̄_{+}^{ϵ}U] ≠ 0.

On the other hand, it is false that we have N^{∞} [S̄_{+}^{ϵ}U] ≠ 0 in the case (ii). (Of course, we still have N^{∞} [uS̄^{ϵ}] ≠ 0.) The criterion in H^{∞} should involve a different definition of long families, probably the following:

∑_{l∈BM(γ)} \frac{|l|^{2}}{δ(l)^{3/2}} = ∞.  (4.20)

Let us state a partial result in the language of S^{(2)}. Note that under the square root transformation, the condition (4.20) becomes

∑_{l∈BM(γ)} \frac{|l|^{2}}{δ(l)} = ∞.

We get the exactly this condition if we apply the non-rigorous argument preceeding the proof of Theorem 4.4.

Proposition. Let M = S^{(2)} and U = e^{iγ}. Suppose

γ′(x) ≥ −C|x|, x → ±∞.

(i) If γ(x) ↛ ±∞ as x → ∓∞, or if γ(∓∞) = ±∞ but

∑_{l∈BM(γ)} \frac{|l|^{2}}{δ(l)(1 + log^{+} |l|)} = ∞,

then

∀ϵ > 0, N^{+} [M^{ϵ}U] = 0.

(ii) If γ(+∞) = −∞ and

∑_{BM(γ)} \frac{|l|^{2} log^{+} |l|}{δ(l)} < ∞,

then

∀ϵ > 0, N^{+} [M̄^{ϵ}U] ≠ 0.

Example. Let L be the harmonic oscillator considered in Example 4.9, and let Λ ⊂ σ(L). The first part of the last proposition provides a sufficient condition for the fact that q on ℝ_{−} and the spectral measure on Λ determine L. This condition is less precise than the one stated in Example 4.9 because we have a larger gap, which characterizes the singularity at infinity rather than the extra length of the known part of the spectrum. On the other hand, the condition in the above proposition does not require any extra assumptions on Λ, cf. (4.18).

In conclusion, we mention that the study of gap functions which are less regular than the standard ones (or even arbitrary) is an interesting and probably difficult problem.

References

[1] Aleksandrov, A. B. Inner functions and related spaces of pseudo continuable functions, Proceedings of LOMI Seminars 170 (1989), 7-33 (Russian)

[2] Aleksandrov, A. B. On A-integrability of the boundary values of harmonic functions, Mat. Zametki 39 (1982), 59–72 (Russian)

[3] Avdonin, S.A., Ivanov, S.A. Families of exponentials. Cambridge Univ. Press, Cambridge, 1995

[4] Beurling, A., Malliavin, P. On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291–302

[5] Beurling, A., Malliavin, P. On the closure of characters and the zeros of entire functions, Acta Math. 118 (1967), 79-93

[6] Boas, R.P. Entire functions. Academic Press, New York, 1954

[7] Borg, G. Uniqueness theorems in the spectral theory of y′′ + (λ − q(x))y = 0, Proc. 11-th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, 276-287

[8] Böttcher A., Silbermann B. Analysis of Toeplitz operators. Academie-Verlag and Springer-Verlag, Berlin, 1990

[9] Bourgain, J. A problem of Douglas and Rudin on factorization, Pacific J. Math. 121 (1986), 47–50

[10] Clark, D. One dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972), 169-91.

[11] Coifman, R.R., Weiss, G. Extensions of Hardy spaces and their use in analysis, Bull. AMS 83 (1977), 569–645

[12] De Branges, L. Hilbert spaces of entire functions. Prentice-Hall, Englewood Cliffs, NJ, 1968 [13] Dyakonov, K. Zero sets and multiplier theorems for star-invariant subspaces, J. Anal. Math.

86 (2002), 247–269

[14] Dym H, McKean H.P. Gaussian processes, function theory and the inverse spectral problem Academic Press, New York, 1976

[15] Everitt, W. N. On a property of the m-coefficient of a second order linear differential equation, J. London Math. Soc. 4 (1972), 443–457

[16] Garnett, J. Bounded analytic functions. Academic Press, New York, 1981

[17] Gesztezy F., Simon B. Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, Trans. AMS 352 (2000), 2765–2787

[18] Gesztezy F., Simon B. m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. d’Analyse Math. 73 (1997), 267-297

[19] Havin, V. P., Jöricke, B. The uncertainty principle in harmonic analysis. Springer-Verlag, Berlin, 1994.

[20] Higgins, J.R. Completeness and basis properties of sets of special functions. Cambridge Univ. Press, Cambridge, 1977 [21] Hochstadt, H., Lieberman, B. An inverse Sturm-Liouville problem with mixed given data,

SIAM J. Appl. Math. 34 (1978), 676–680

[22] Horváth M. Inverse spectral problems and closed exponential systems, Preprint (2004)

[23] Hruschev S. V., Nikolskii, N. K., Pavlov, B. S. Unconditional bases of exponentials and of reproducing kernels, Lecture Notes in Math., Vol. 864, 214–335

[24] Khodakovsky, A. M. Inverse spectral problem with partial information on the potential. PhD Thesis, Caltech, 1999

[25] P. Koosis, The logarithmic integral, Vol. I & II. Cambridge Univ. Press, Cambridge, 1988 [26] Krein, M. On the theory of entire functions of exponential type, Izv. AN SSSR 11 (1947),

309–326 (Russian)

[27] Lax P. D., Phillips R.S. Scattering theory. Academic Press, New York, 1967

[28] Levin, B. Distribution of zeros of entire functions, AMS, Providence, RI, 1980

[29] Levinson, N. Gap and density theorems. AMS, New York, 1940

[30] Levitan, B.M., Sargsjan, I.S. Sturm-Liouville and Dirac operators. Kluwer, Dordrecht, 1991

[31] Marchenko, V. Some questions in the theory of one-dimensional linear differential operators of the second order, I, Trudy Mosk. Mat. Obsch. 1 (1952), 327-420.

[32] Marchenko, V. Sturm-Liouville operators and applications. Birkhauser, Basel, 1986

[33] Paley, R., Wiener, N. Fourier transform in the complex domains. AMS, New York, 1934 [34] Nikolskii, N. K. Treatise on the shift operator. Springer-Verlag, Berlin-New York, 1986 [35] Nikolskii, N.K. Operators, functions, and systems: an easy reading, Vol. I & II. AMS,

Providence, RI, 2002

[36] Sarason, D. Sub-Hardy Hilbert spaces in the unit disk. Univ. of Arkansas Lecture Notes in the Math. Sciences, vol. 10, J. Wiley and Sons, New York, 1994

[37] Sarason, D. Kernels of Toeplitz operators, Oper. Theory Adv. Appl. 71 (1994), 153–164 [38] Redheffer, R. Completeness of sets of complex exponentials, Advances in Math. 24, 1977,

1-62

[39] Titchmarsh, E.C. Eigenfunction expansion associated with second order differential equations , Clarendon Press, Oxford, 1946

[40] Treil S., Volberg, A. Embedding theorems for invariant subspaces of inverse shift, Proceedings of LOMI Seminars, 149 (1986), 38–51

[41] Young, R. M. An introduction to non-harmonic Fourier series. Academic Press, New York, 1980

[42] Zygmund, A. Trigonometric series, Vol. I & II. Cambridge University Press, Cambridge, 1959

California Institute of Technology, Department of Mathematics, Pasadena, CA 91125, USA

   E-mail address: makarov@its.caltech.edu

Texas A& M University, Department of Mathematics, College Station, TX 77843, USA

   E-mail address: alexeip@math.tamu.edu