Killip-Simon problem on two disjoint intervals

Killip-Simon problem on two disjoint intervals

Benjamin Eichinger

Institute of Analysis Johannes Kepler University Linz

July 20, 2015

Killip-Simon theorem

Theorem

Letdσ(x) =σ⁰(x)dx+dσ_sing(x) be the spectral measure of a one-sided Jacobi matrixJ₊. Then the following are equivalent:

(op) J+−J^◦+ ∈HS

(sp) The spectral measure dσ is supported on X∪[−2,2]and Z 2

−2

|logσ⁰(x)|p

4−x²dx+ X

xk∈X

q x_k²−4

3

<∞.

Isospectral torus

LetE = [b₀,a₀]\Sg

j=1(a_j,b_j)and for a two-sided Jacobi matrixJ we define

r±(z;J) =D

(J±−z)⁻¹e^−1±1

2 ,e^−1±1

2

E .

Definition

A Jacobi matrixJ is called reflectionlesson E if

a₀²r+(x+i0) = 1

r−(x−i0) for almost allx ∈E. The classJ(E)is formed by Jacobi matrices, which are reflectionless on their spectral setE.

Spectral theory of periodic Jacobi matrices

Theorem

LetE = [b₀,a₀]\Sg

j=1(a_j,b_j).ThenE is the spectrum of a p periodic two-sided Jacobi matrixJ^◦ if and only of there exists a polynomialT_p(z)such thatT_p⁻¹([−2,2]) =E with some additional properties.

Theorem (Magic formula) LetE = [b₀,a₀]\Sg

j=1(a_j,b_j)be the spectrum of a p periodic Jacobi matrix. Then

◦

J ∈J(E)if and only if Tp(

◦

J) =S^−p+S^p.

Damanik-Killip-Simon theorem

Theorem (Damanik-Killip-Simon theorem) The following are equivalent:

(op) T(J+)−(S₊^p+ (S₊^∗)^p)∈HS

(sp) The spectral measure dσ is supported on X∪E and Z

E

|logσ⁰(x)|p

dist(x,R\E)dx+ X

xk∈X

pdist(x_k,E)³<∞.

Why does this Magic Formula only exist in the periodic case?

What can we do in the general case?

The Riemann surface R

From now on, we assume thatE = [b₀,a₀]\(a₁,b₁).

We define

s(z) = (z−a₀)(z−b₀)(z−a₁)(z−b₁)

and

R_E ={(P = (z,w) : w² =s(z)}∪{∞₊,∞₋} the compactRiemann surfaceof

s(z).

It is convenient to imagine R_E as a two-sheeted Riemann sur- face, where p

s(z) > 0, for z >a0 on R₊.

a0

b0 a1 b1

a₀

b₀ a₁ b₁

ℛ+

ℛ-

⊕

⊖

⊕

⊕

⊕

⊖

⊖ ⊖

The isospectral torus

By the reflectionless property we extendr₊ to a meromorphic function onR_E:

a²₀r+(P) =

(a²₀r₊(z) ifP = (z,w)∈ R₊

1

r−(z) ifP = (z,w)∈ R₋. Let us define

D_E ={(x, ) : x ∈[a₁,b₁], =±1}.

One can show that for anyJ ∈J(E) we have

a²₀r+((z,w)) = 1

2(w +w0

z −x0

−(z−q0)),

where(x0, 0)∈ D_E,q0=q0((x0, 0))∈Randw0 =w0(x0, 0).

Moreover, each such a function definesJ ∈J(E).

The isospectral torus, Abel map

J(E)_aa ^{oo r //}

!!

D_E

>>

~~ A

T

a0

b0 a1 b1

(x0,1)

(x0,-1)

S₊

The Abel map

There exists a conformal map fromR_E onto a rectangle with vertices ^−1−τ₂ ,^1−τ₂ ,^1+τ₂ ,^−1+τ₂ inC. It is given by

u(z) =C

z

Z

a0

dz ps(z),

x0

x1 S+

S-

u∞+

u∞-

u0

1/2 uc

τ/2

-1/2

-τ/2

The Abel map

u∞+

u∞-

u∞+

u∞-

We call the map

A:R_E →C/L z 7→u(z),

whereL=Z+τZ, the Abel map.

Meromorphic functions on R

and elliptic functions

Definition

Meromorphic functions onC with periods 1 andτ bear the name elliptic functions.

Theorem

The properly counted (i.e., counted with multiplicity taking into account) number of poles of a nonconstant elliptic functionf(u) in a period parallelogram is equal to the properly counted number of zeros.

Theorem

Let{α_k}^N_k=1,{β_k}^N_k=1 are a system of points in a period parallelogram. Thenα_k are the zeros on β_k are the poles of an elliptic function, if and only if

N

P

k=1

α_k − P^N

k=1

β_k ∈L.

Meromorphic functions on R

and elliptic functions

u∞+

u∞-

u0

uc

u1 τ/2

-1/2 1/2

-τ/2

one-to-one correspondence between meromorphic functions on R_E and elliptic functions w.r.t. L.

r₊, meromorphic function on R_E with poles at {∞₋,x0}and zeros at {∞₊,x1}.

u∞+ +u1−(u∞−+u0)∈Z

Shift on J(E ) and on T

One can show that

r+(z,w) =− 1

b₀−z+r₊⁽¹⁾(z,w). Thus, the zero of r₊ is the pole ofr₊⁽¹⁾.

Let µ=u∞+ −u∞−. Thenu1 =u0−µ mod Z.

J(E)_aa ^{oo r+ //}

!!

D_E

>>

~~ A

T

μ

u₀ u1

Functional models for Jacobi matrices

Theorem (Jacobi theta function)

There exists an analytic functionθ₁ onC with the following properties:

θ1(u±1) =−θ₁(u)

θ₁(u±τ) =−e^−iπτe^∓2πiuθ₁(u) Moreover, the zeros ofθ₁ are given by

v =m+nτ, m,n ∈Z.

Recall, thatr+ is a meromorphic functions onR_E with poles at {∞₋,x0}and zeros at {∞₊,x1}. Thus, it is given by

r+(z,w) =Cθ₁(u−u∞+)θ₁(u−u₁) θ1(u−u∞−)θ1(u−u0).

Functional models for Jacobi matrices

Now we set

B(u) = θ1(u−u∞₊) θ1(u−u∞−) and

k^α(u) =C θ1(u−u0)

θ₁(u−u_c−), K^α(u) = k^α(u) pk^α(u∞+), wherec−=x−+i^κ₂ is a certain normalization point.

This functions are no elliptic functions! We have

B(u+1) =B(u) and K^α(u+1) =K^α(u), and

B(u+τ) = θ1(u−u∞++τ)

θ₁(u−u∞−+τ) = θ1(u−u∞+) θ₁(u−u∞−)

e^{−i(u−u∞+)} e^{−i(u−u∞−)}. Thus, we see that

B(u+τ) =e^iµB(u).

Functional models for Jacobi matrices

Similarly, we can show that

K^α(u+τ) =e^iαK^α(u), whereα=u0−u_uc−. In general, we define

K^α−kµ(u) = θ1(u−(u0−kµ)) θ₁(u−u_c−) . Again, we see that

K^α−kµ(u+τ) =e^i(α−kµ)K^α−kµ(u).

The space H

(α)

LetΩ =C\E. For α∈T, we define H²(α) as the space of multivalued, analyticfunctions f^α(z) inΩ, which satisfy:

|f^α|² has a harmonic majorant in Ω,

g^α(u) =f^α(z), satisfiesg^α(u+τ) =e^iαg^α(u).

Note, that|f^α|² is single-valued. The norm is given by the value at infinity of the least harmonic majorant of|f^α|². The corresponding scalar product is given by

hf₁^α,f₂^αi= Z

E

f₁^α(x)f₂^α(x)ω(dx,∞),

whereω(dx,∞) denotes the harmonic measure of infinity of the domainΩ.

It is given by

ω(dx,∞) = i 2π

x−c ps(z).

Functional models for Jacobi matrices

Theorem Let

e_n^α(z) =Bⁿ(z)K^α−kµ(z) Then

enn ∈N is a ONB of H²(α) e_nn ∈Z is a ONB of L²(α)

Theorem

Lete_n^α be defined as above, then the multiplication operatorz in L²(α) with respect to the basis{e_n^α}is a Jacobi matrix J =J(α):

ze_n^α =a_n(α)e_n−1^α +b_n(α)e_n^α+an+1(α)e_n+1^α ,

Sketch of the proof

We only sketch, why the multiplication byz is a Jacobi matrix. We have

r+(z) =− 1

z −b₀+a₁²r⁽¹⁾(z) =Cθ₁(u−u∞+)θ₁(u−u₁) θ1(u−u∞−)θ1(u−u0). That is,

a0r+ =−BK^α−µ

K^α and a1r₊⁽¹⁾=−BK^α−2µ K^α−µ. Thus we have

a₀K^α

BK^α−µ = (z −b₀)−a₁BK^α−2µ K^α−µ ⇔ zBK^α−µ=a0K^α+b0BK^α−µ+a1B²K^α−2µ⇔ ze₁^α =a0e₀^α+b0e₁^α+a1e₂^α

Proof of the magic formula

S⁻²JS =J ⇔2µ∈Z. This is only a property of the spectrum not ofJ! Consider, in this case the basis{e_n^α} has the following form

{. . . ,B⁻¹K^α+µ,K^α,BK^α+µ,B²K^α, . . .}.

Recallµ=u∞+ −u∞−. There exists a function ψ(z,w) =

θ1(u−u∞+) θ1(u−u∞−)

2

=B(u)², on R_E. ψ+ψ⁻¹ is a function on Ω =C\E. Thus,

ψ(z,w) + 1

ψ(z,w) =T2(z).

This is the magic formula.

`² e_n S² J(α) L²(α) e_n^α ψ z

Functional models in the general case

Now let us consider the case 2µ /∈Z. In this case we define

ψ= θ1(u−u∞+)θ1(u−uc+)

θ₁(u−u∞−)θ₁(u−u_c−) =B(u)B_c(u), whereµ+µc ∈Z. In this case we have

ψ(z,w) + 1

ψ(z,w) =λ0z +d0+ λ1

d₁−z = ∆(z).

Functional models in the general case

Letk^α(z,c) =k_c^α(z) be the reproducing kernels of the space H²(α). We defineK_ψ(α) =H²(α) ψH²(α). K_ψ is spanned by:

f₀^α(z) =λ(α) k_d^α

1(z) kk_d^α

1(z)k and f₁^α(z) =B_d1(z)k^α+µ(z) kk^αµk , Thus,H²(α) =Kψ(α)⊕ψKψ(α) +ψ²Kψ(α)⊕. . . .

Theorem

The system of functions

f_n^α =

(ψ^mf₀^α, n=2m ψ^mf₁^α, n=2m+1

forms an orthonormal basis in H²(α) forn ∈Nand forms an orthonormal basis in L²(α) forn ∈Z.

Functional models in the general case, SMP-matrices

Definition

We call the operatorA(α)∈L(`²), which corresponds to the multiplication byz in this basis a SMP-matrix.

A(α) is two periodic. σ(A(α)) =E

A=























. ..

. .. ... A A^∗ B A

A^∗ B A A^∗ . .. ...

. ..





















 ,

A=

0 0 p0 p1

, B=

p0q0+d0 p1q0

p1q0 p1q1

.

Byconstruction, we obtain the magic formula for SMP matrices.

`² e_n S² A(α) L²(α) f_n^α ψ z

Theorem

LetA(E) be the set of all SMP matrices of period two with their spectrum on E. Then we have:

A∈A(E) ⇔ ∆(A) =λ0A+d0+λ1(A−d1)⁻¹ =S²+S⁻²

The Jacobi flow on SMP-matrices

Definition

We define the mapF : A(E)→J(E), by F(A(α)) =J(α).

Lemma

LetkP₊A(α)e−1k² and{a_k,b_k}be the Jacobi parameters of J(α).

Then we have

p1q1=b−1, p²₀+q₀²=a₀² It follows immediately, sincee₋₁^α =f₋₁^α .

The Jacobi flow on SMP-matrices

Definition

The Jacobi flow inA(E)is defined by J(A(α)) =A(α−µ).

A ^{F //}

J

J

S

A ^{F //}J

Explicit formulas for the Jacobi flow

Theorem

LetU(α)be the periodic 2×2-block diagonal unitary matrix given by

U(α)

e2m e2m+1

=

e2m e2m+1

u(α), (1) where

u(α) = 1

q

p²₀(α) +p₁²(α)

p₀(α) p₁(α) p1(α) −p₀(α)

(2)

Then

JA(α) :=A(µ⁻¹α) =S⁻¹U(α)^∗A(α)U(α)S. (3)