Families of spectral measures with mixed types 2: New proofs.

R. del Rio, ^{∗} S. Fuentes ^{†} A. Poltoratski ^{‡}

Introduction.

In this note we shall continue the study of the behavior of families of measures μ_{λ} which depend on a real parameter λ. The measures we are interested in, will be spectral measures of self adjoint operators. Let A be a cyclic selfadjoint operator, φ its cyclic vector and μ the corresponding spectral function. The μ_{λ} will be the spectral measures associated to the operators

A_{λ} = A + λ(·, φ)φ

In the same setting we can treat families of spectral measures μ_{θ} associated to Sturm-Liouville operators H_{θ} generated by

H_{θ}u = −u′′ + q(x)u 0 ≤ x < ∞

where q is real valued, locally integrable function, and the boundary condition

u(0) cos θ + u′(0) sin θ = 0 0 ≤ θ < π

The measures mentioned above appear in the integral representation of analytic functions which take the upper half plane into itself.

∗IIMAS-UNAM, Circuito Escolar, Ciudad Universitaria, 04510, México, D.F., México. E-mail: delrio@servidor.unam.mx

†

‡Department of Mathematics, College Station, Texas 77843–3368, U.S.A. Email: alexei.poltoratski@math.tamu.edu

Families of spectral measures with mixed types 2: New proofs. 2

We shall call these “Pick functions”, following [6]. This function will play an essential role in what follows.

A basic problem in spectral theory of selfadjoint operators is to understand the relation between the operators and their spectra. Since all the information about spectra is contained in the spectral measures, it is of importance to understand how different features of these measures are related to the corresponding operators.

In particular if the operator is changed in some way, for example if we add a rank one perturbation or if we modify the boundary condition, then we are interested to know which spectral properties will be stable, which ones will be more sensitive, what is possible, and which situations will not occur.

Of particular interest to us in this note is the study of mixed situations. We address the following questions: which sets B ⊂ ℝ are such that for all θ ∈ B the measures μ_{θ} may have absolutely continuous and singular parts coexisting? Which kinds of singular parts (continuous, pure point) allow this coexistence? Regarding the first question, it was shown in [13]that B could be all the interval (−π/2, π/2]. We shall show here that it is possible to have coexistence just in the interval (−π/2, 0). Concerning the second question we shall see that virtually any kind of singular spectrum will allow coexistence.

1 Preliminaries

Every analytic function F (z) which takes the upper half plane into itself has an integral representation of the form

F (z) = a + b + z _{ℝ}^{∫} [\frac{1}{t − z} − \frac{t}{1 + t^{2}} ] dμ(t)  (1)

where a, b ∈ ℝ, b ≥ 0 and μ is a non negative Borel measure which satisfies

_{ℝ}^{∫} \frac{dμ(t)}{1 + t^{2}} < ∞

and conversely, any function of this form is analytic and takes the upper half plane into itself, see [6] and [11]

When μ generates a family of spectral measures for Sturm-Liouville operators in the half axis, we will have b = 0 in the expression (1) and in case of spectral measures for rank one perturbations we will have

F (z) = _{ℝ}^{∫} \frac{dμ(t)}{t − z}

where μ is a finite measure. Consider the family of Pick functions

f_{θ}(z) := \frac{cos θ + z sin θ}{sin θ − z cos θ}  (2)

Let K(z) be a Pick function such that 0 ≤ ImK(z) ≤ π.

Define

K_{μ_{β}}(z) := f_{β}(K(z))
K_{ν_{α}}(z) := f_{α}(exp K(z))

Where μ_{β} and ν_{α} denote the measures associated to K_{μ_{β}} and K_{ν_{α}} respectively according to the integral representation theorem for Pick functions. Observe that K(z) = K_{μ_{π/2}}(z). We shall study the relation between the singular parts of the measures μ_{β} and ν_{α}, denoted by μ_{β}^{s} and ν_{α}^{s} respectively. The next theorem gives a very explicit result

Theorem 1 Let ν_{α} and μ_{β} be defined as above and assume that

α(β) = tg^{−1}(exp tgβ), β ∈ (−π/2, π/2)

then

μ_{β}^{s}(E) = α′(β) ν_{α}^{s}(E)

for every Borel set E Remark 1

This result was proven in [13] and it is true if instead of exp(z) in the definition of K_{ν_{α}}(z) we use any other function analytic in the neighborhood of the strip S = {0 < Im z < π} which takes S to the upper half–plane and ℝ to ℝ. The definition of α(β) in the statement would have to be changed accordingly.

2 Coexistence of spectra

We now show how to construct famlies of measures which have singular and absolutely continuous parts. The following well known observations will be useful in the sequel. We use the symbol γ^{ac} to denote the absolutely continuous part of a measure γ.

Proposition 1 Let J be an open interval

a) The function K_{μ}(z) can be continued analyticaly accross J if and only if μ(J) = 0

b) μ^{ac}(J) > 0 ⟺ μ_{β}^{ac}(J) > 0 for every β ∈ (−π/2, π/2)

c) μ(J′) > 0 for every subinterval J′ ⊂ J if and only if μ_{β}(J′) > 0 for every subinterval J′ ⊂ J, β ∈ (−π/2, π/2).

For the proof of a) we refer to [7, p. 32 ] for b) see [12, theorem 2.1 ] and for c) see [7, p. 38, theorem 2.52 ].

Consider a fixed interval I and take an arbitrary measure ν_{π/2} which is purely singular in I, i.e. ν_{π/2}^{s}(J) > 0 for every interval J ⊂ I and ν_{π/2}^{ac} = 0. Let K_{μ_{π/2}} be defined as

K_{μ_{π/2}}(z) = log(K_{ν_{π/2}}(z))  (3)

Set as above

K_{μ_{β}}(z) = f_{β}(K_{μ_{π/2}}(z))

Since log is a Pick function K_{μ_{π/2}} is a Pick function too.

Theorem 2 For every subinterval J ⊂ I we have:

a) μ_{β}^{s}(J) > 0, β ∈ (−π/2, π/2)

b) μ_{β}^{ac}(J) > 0, β ∈ (−π/2, π/2]

Proof

a) From the stability of the absolutely continous part, Prop. 1 b), it follows that ν_{α} is purely singular in I for α ∈ (−π/2, π/2]

Since

K_{ν_{α}}(z) = f_{α}(exp K_{μ_{π/2}}(z))

we can apply Theorem 1 to get

μ_{β}^{s}(J) = α′(β)ν_{α}^{s}(J), J ⊂ I
for every β ∈ (−π/2, π/2)
α ∈ (0, π/2)

and therefore μ_{β}^{s}(J) > 0.

b) If the function log is determined in such a way that it is real on the positive half axis then

ImK_{μ_{π/2}}(x + i0) = Im log K_{ν_{π/2}}(x + i0)

is bounded by π and it follows that μ_{π/2} is absolutely continuous, since the support of its singular part is the set, see [8],

{x/K_{μ_{π/2}}(x + i0) = ∞}

If there is an interval J ⊂ I such that μ_{π/2}(J) = 0 then from Proposition 1 a) we know that K_{μ_{π/2}}(z) can be continued analyticaly accross J and from (3) it follows that K_{ν_{π/2}}(z) too. This implies that ν_{π/2}(J) = 0 giving us a contradiction. Therefore μ_{π/2}(J) > 0 for every J ⊂ I and from the stability of the absolutely continuous part Prop. 1 b) we have μ_{β}^{ac}(J) > 0, for β ∈ (−π/2, π/2].

q.e.d.

Therefore, given an arbitrary family of purely singular measures {ν_{α}} it is posible to construct a family {μ_{β}} with equivalent singular parts, but with non-zero absolutely continuous components. This completes results of [4] where it was shown that coexistence of a.c. and singular parts was possible just for a set of β′s of positive Lebesgue measure.

Consider a fixed interval I and take a set E ⊂ I such that for every subinterval J ⊂ I we have

0 < |E ∩ J| < |J|

Such sets can be constructed. See [4] and [5, examples 4 and 5] Let

u(x) = \{ \begin{matrix}1 & x ∈ E \\ 0 & x ∈ E^{c} \end{matrix}

and define

K_{μ_{π/2}}(z) = ∫ ; ℝ \frac{d μ_{π/2}(λ)}{λ − z}

where

d μ_{π/2} = u dx

and set μ_{π/2} outside I such that

∫ ; ℝ \frac{dμ_{π/2}(λ)}{λ^{2} + 1} < ∞

holds, or with the necessary decay required by the Gelfand-Levitan inverse theorem in order to have a Sturm-Liouville problem. See [9].

It was shown in [PK] that for the family of measures μ_{β} the following holds:

Theorem 3 For every subinterval J ⊂ I we have

a) μ_{β}^{s}(J) > 0 for all β ∈ (−π/2, π/2)

b) μ_{β}^{ac}(J) > 0 for all β ∈ (−π/2, π/2]

In the next theorem we shall see that if we multiply de measure μ_{π/2} used in Theorem 3 by a constant less than 1, then we get a singular component only for half of the coupling constants β.

Let d μ_{π/2} = u(x)dx as in (3)

Define

K_{γ_{α}}(z) := f_{α}(exp(aK_{μ_{π/2}}(z)))

where 0 < a < 1

Theorem 4 For the family of measures γ_{α} we have γ_{α}^{s}(J) > 0 for every subinterval J ⊂ I if and only if α ∈ (0, π/2). If α ∈ (−π/2, 0] then γ_{α} has only an absolutely continuous part.

Proof

If we take in Theorem 1 instead or exp, the function F (z) = az, (see Remark 1) and define

K_{μ_{β}^{∗}}(z) = f_{β}(aK_{μ_{π/2}}(z))

then we get according to Theorem 1

μ_{β}^{s}(E) = α′(β) μ_{α}^{∗s}(E)

where α(β) = tg^{−1}(a tgβ) β, α ∈ (−π/2, π/2)

Applying Theorem 2 a) it follows that μ_{α}^{∗s}(J) > 0 for every subinterval J ⊂ I α ∈ (−π/2, π/2)

Now consider

K_{γ_{α}}(z) := f_{α}(exp K_{μ_{π/2}^{∗}}(z))  (4)

Appling Theorem 1 we get

μ_{β}^{∗s}(E) = α′(β) γ_{α}^{s}(E)

for every Borel set E and

α(β) = tg^{−1}(exp tgβ)
β ∈ (−π/2, π/2)
α ∈ (0, π/2)

Therefore

γ_{α}^{s}(J) > 0 for every subinterval J ⊂ I

and for every α ∈ (0, π/2)

To see that γ_{α}^{s} = 0 for α ∈ (−π/2, 0], let us recall that a minimal support for γ_{α}^{s} is given by the set

A := {x/K_{γ_{α}}(x + i0) = ∞}

see [8], for example which by the definition of K_{α} through (4) and using (2) can be seen to be equal to the set

{x/K_{γ_{π/2}}(x + i0) = tgα}

If we assume that γ_{α}^{s} ≠ 0 then A ≠ ∅.

Now from (4) we have, where | · | denotes Lebesgue measure

K_{γ_{π/2}}(x + i0) = exp (∫_{E} \frac{a}{λ − (x + i0)}dλ)
= exp (Re + iπa lim_{ε↓0} \frac{|E ∩ (x − ε, x + ε)|}{2ε})

where Re is the real part of K_{μ_{π/2}^{∗}}(x+i0), see [10] page 44 for example.

Since for x ∈ A the right side of the last equation has to be tgα ∈ ℝ we should have

lim_{ε↓0} \frac{|E ∩ (x − ε, x + ε)|}{2ε} = 0 or \frac{1}{a}

since \frac{|E ∩ (x − ε, x + ε)|}{2ε} ≤ \frac{2ε}{2ε} = 1 and a < 1 we have

lim_{ε↓0} \frac{|E ∩ (x − ε, x + ε)|}{2ε} = 0

i.e.

tgα = exp(Re) > 0
Therefore α ∈ (0, π/2)
Hence if α ∈ (−π/2, 0] then γ_{α}^{s} = 0

Remark 4

In the case of a family of ergodic operators H_{ω} we can not have an analogous situation to the one described above taking instead of θ the parameter ω. This follows from a theorem of KMS see [1] which tell us that there exists sets ∑_{s} and ∑_{a c} such that

σ_{s}(H_{ω}) = _{∑s}
σ_{a c}(H_{ω}) = _{∑a c}

for a.e.ω

Open Problem: Prove that given any set B ⊂ (−π/2, π/2] it is possible to find a family of measures μ_{β} s.t. μ_{β}^{s}(J) > 0 ⇔ β ∈ B and μ_{β}^{ac}(J) > 0 ∀β ∈ (−π/2, π/2], for every interval J ⊂ I.

Next we shall show how to construct spectral families of measures having singular and a.c. parts, without using Theorem 1.

Let us consider the Pick functions

f_{λ}(z) = \frac{1}{λ + \frac{1}{z}} , λ ∈ ℝ

and use the notation

K_{γ}(z) := ∫ \frac{dγ(x)}{x − z}  (5)
K_{γ_{λ}}(z) := f_{λ}(K_{γ}(z))  (6)
λ ∈ ℝ

when (6) holds, we will say that {γ_{λ}}_{λ∈ℝ} is the family generated by γ. Notice that γ_{0} = γ

The following result, which is essentially Lemma 5.3 of [2], will be useful

Lemma 1

η(A) = _{ℝ}^{∫} μ_{λ}(A)dν(λ)  (7)

if and only if

K_{η}(z) = K_{ν}(−K_{μ}^{−1}(z))  (8)

for every Borel set A.

Proof

⇒) Lemma 5.3 of [2]

⇐) By the definition of μ_{λ}, (7) implies

_{ℝ}^{∫} \frac{dη(x)}{x − z} = _{ℝ}^{∫} dν(λ)_{ℝ}^{∫} \frac{dμ_{λ}(x)}{x − z}

Using Stone Weierstrass for the algebra of finite sums of functions of the form (x − z)^{−1} we get (7)

q.e.d.

Proposition 2 Define

η(A) = _{ℝ}^{∫} μ_{ξ}(A)dν(ξ)  (9)

then

η_{λ}(A) = _{ℝ}^{∫} μ_{ξ}(A)dν_{λ}(ξ)

for every Borel set A, λ ∈ ℝ, where η_{λ} and ν_{λ} are defined as in (6).

Proof Definition (9) implies K_{η} = K_{ν}(−K_{μ}^{−1}) by Lemma 1. Therefore

f_{λ}(K_{η}(z)) = f_{λ}(K_{ν}(−K_{μ}^{−1}(z)))

i.e.

K_{η_{λ}}(z) = K_{ν_{λ}}(−K_{μ}^{−1}(z))

and applying Lemma 1 again we obtain

η_{λ}(A) = ∫ μ_{ξ}(A)dν_{λ}(ξ)

for every Borel set A, λ ∈ ℝ

q.e.d.

Construct now two measures μ and ν as follows. Let μ be purely singular in a fixed interval I, i.e. μ = μ^{s} and μ(J) > 0 for every subinterval J ⊂ I and define ν as

dν = u(x)dx

where

u(x) = \{ \begin{matrix}1 & x ∈ [a, b] \\ 0 & otherwise \end{matrix}

Theorem 5 Let {μ_{ξ}}_{ξ∈ℝ} and {ν_{λ}}_{λ∈ℝ} be the families generated by the measures μ and ν defined above and set

η(λ, A) := ∫ μ_{ξ}(A)dν_{λ}(ξ)

then

a) η_{λ}(A) = η(λ, A) η_{λ} as defined by (6)

b) η_{λ}^{ac}(J) > 0 for every subinterval J ⊂ I

c) η_{λ}^{s}(J) > 0 for every subinterval J ⊂ I

Proof Part a) follows from Proposition 2. The function

K_{ν}(z) = _{ℝ}^{∫} \frac{dν(x)}{x − z} = ∫_{a}^{b} \frac{dx}{x − z}

has an analytic extension accross ℝ\[a, b] and takes every positive real value in (−∞, a) and every negative value in (b, ∞) only once. Therefore K_{ν_{λ}} has a pole at the point ξ_{0} where K_{ν}(ξ_{0}) = −\frac{1}{λ} and this pole is a point mass of ν_{λ}. We can write then

η_{λ}(A) = ∫ μ_{ξ}(A)dν_{λ}^{ac}(ξ) + μ_{ξ_{0}}(A)ν_{λ}(ξ_{0})

Part c) follows immediately since μ_{ξ_{0}} is purely singular by construction. To prove b) assume that | A |= 0. Then μ_{ξ}(A) > 0 only for a set of ξ′s of Lebesgue measure zero. This can be seen using the formula

_{ℝ}^{∫} μ_{ξ}(A)dξ =| A |

see [12]. Since ν_{λ}^{ac} is absolutely continuous then

∫ μ_{ξ}(A)dν_{λ}^{ac}(ξ) = 0

and therefore η_{λ}^{ac} ≠ 0.

¿From the construction of {μ_{ξ}} we know that μ_{ξ}(J) > 0 for every subinterval J ⊂ I.

Hence

∫ μ_{ξ}(J)dν(ξ) = ∫_{a}^{b} μ_{ξ}(J)dξ > 0

and

η_{0}^{ac}(J) > 0

¿From the stability of the a.c. part we get η_{λ}^{ac}(J) > 0 for every λ ∈ ℝ

q.e.d.

If we want a family of measures generated by f_{θ} as defined in (2) instead of f_{λ} as defined in (6) we can just use the relation

dγ_{θ} = (1 + λ^{2})dη_{λ}
e^{iθ} = \frac{1 + iλ}{1 − iλ}, λ ∈ ℝ

as shown in [3, Section 1].

Acknowledgment This work was partially supported by projects IN-102998 PAPIIT-

UNAM and 27487E CONACyT.

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Families of spectral measures with mixed types 2: New proofs. 15

http//www.ma.utexas.edu/mp arc-bin/mpa?yn=00-489. To appear in Proceedings of the Kovalewski Conference.