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Killip-Simon problem on two disjoint intervals Benjamin Eichinger Institute of Analysis Johannes Kepler University Linz July 20, 2015 B. Eichinger Killip-Simon problem on two disjoint intervals Killip-Simon theorem Theorem Let dσ(x) = σ 0 (x)dx + dσsing (x) be the spectral measure of a one-sided Jacobi matrix J+ . Then the following are equivalent: ◦ (op) J+ − J+ ∈ HS (sp) The spectral measure dσ is supported on X ∪ [−2, 2] and Z 2 p 3 Xq | log σ 0 (x)| 4 − x 2 dx + xk2 − 4 < ∞. −2 xk ∈X B. Eichinger Killip-Simon problem on two disjoint intervals Isospectral torus S Let E = [b0 , a0 ] \ gj=1 (aj , bj ) and for a two-sided Jacobi matrix J we define E D r± (z; J) = (J± − z)−1 e −1±1 , e −1±1 . 2 2 Definition A Jacobi matrix J is called reflectionless on E if a02 r+ (x + i0) = 1 for almost all x ∈ E . r− (x − i0) The class J(E ) is formed by Jacobi matrices, which are reflectionless on their spectral set E . B. Eichinger Killip-Simon problem on two disjoint intervals Spectral theory of periodic Jacobi matrices Theorem Let E = [b0 , a0 ] \ Sg j=1 (aj , bj ). Then E is the spectrum of a p ◦ periodic two-sided Jacobi matrix J if and only of there exists a polynomial Tp (z) such that Tp−1 ([−2, 2]) = E with some additional properties. Theorem (Magic formula) S Let E = [b0 , a0 ] \ gj=1 (aj , bj ) be the spectrum of a p periodic ◦ Jacobi matrix. Then J ∈ J(E ) if and only if ◦ Tp (J) = S −p + S p . B. Eichinger Killip-Simon problem on two disjoint intervals Damanik-Killip-Simon theorem Theorem (Damanik-Killip-Simon theorem) The following are equivalent: p ∗ )p ) ∈ HS T (J+ ) − (S+ + (S+ (op) (sp) The spectral measure dσ is supported on X ∪ E and Z Xp p 3 dist(xk , E ) < ∞. | log σ 0 (x)| dist(x, R \ E )dx + E xk ∈X Why does this Magic Formula only exist in the periodic case? What can we do in the general case? B. Eichinger Killip-Simon problem on two disjoint intervals The Riemann surface RE From now on, we assume that E = [b0 , a0 ] \ (a1 , b1 ). We define ℛ+ s(z) = (z−a0 )(z−b0 )(z−a1 )(z−b1 ) b0 ⊕ ⊖ a1 b1 ⊕ ⊖ a0 a1 b1 ⊖ ⊕ a0 and RE = {(P = (z, w ) : w 2 = s(z)}∪{∞+ , ∞− } the compact Riemann surface of s(z). It is convenient to imagine RE as a two-sheeted p Riemann surs(z) > 0, for face, where z > a0 on R+ . B. Eichinger ℛ- b0 ⊖ ⊕ Killip-Simon problem on two disjoint intervals The isospectral torus By the reflectionless property we extend r+ to a meromorphic function on RE : ( a02 r+ (z) if P = (z, w ) ∈ R+ a02 r+ (P) = 1 if P = (z, w ) ∈ R− . r− (z) Let us define DE = {(x, ) : x ∈ [a1 , b1 ], = ±1}. One can show that for any J ∈ J(E ) we have 1 w + w0 a02 r+ ((z, w )) = ( − (z − q0 )), 2 z − x0 where (x0 , 0 ) ∈ DE , q0 = q0 ((x0 , 0 )) ∈ R and w0 = w0 (x0 , 0 ). Moreover, each such a function defines J ∈ J(E ). B. Eichinger Killip-Simon problem on two disjoint intervals The isospectral torus, Abel map J(E ) o / DE > r a ! A ~ T S+ (x0,1) b0 a1 (x0,-1) B. Eichinger b1 a0 Killip-Simon problem on two disjoint intervals The Abel map There exists a conformal map from RE onto a rectangle with 1−τ 1+τ −1+τ vertices −1−τ in C. It is given by 2 , 2 , 2 , 2 Zz u(z) = C a0 dz p , s(z) τ/2 uc S+ u0 x0 -1/2 u∞ - u∞ + 1/2 S- x1 -τ/2 B. Eichinger Killip-Simon problem on two disjoint intervals The Abel map We call the map u∞- u∞ + A : RE → C/L z 7→ u(z), u∞- u∞ + where L = Z + τ Z, the Abel map. B. Eichinger Killip-Simon problem on two disjoint intervals Meromorphic functions on RE and elliptic functions Definition Meromorphic functions on C with periods 1 and τ bear the name elliptic functions. Theorem The properly counted (i.e., counted with multiplicity taking into account) number of poles of a nonconstant elliptic function f (u) in a period parallelogram is equal to the properly counted number of zeros. Theorem N Let {αk }N k=1 , {βk }k=1 are a system of points in a period parallelogram. Then αk are the zeros on βk are the poles of an N N P P elliptic function, if and only if αk − βk ∈ L. k=1 B. Eichinger k=1 Killip-Simon problem on two disjoint intervals Meromorphic functions on RE and elliptic functions one-to-one correspondence between meromorphic functions on RE and elliptic functions w.r.t. L. τ/2 u1 -1/2 uc u∞- u0 u∞ + 1/2 r+ , meromorphic function on RE with poles at {∞− , x0 } and zeros at {∞+ , x1 }. u∞+ + u1 − (u∞− + u0 ) ∈ Z -τ/2 B. Eichinger Killip-Simon problem on two disjoint intervals Shift on J(E ) and on T One can show that r+ (z, w ) = − 1 (1) . b0 − z + r+ (z, w ) (1) Thus, the zero of r+ is the pole of r+ . Let µ = u∞+ − u∞− . Then u1 = u0 − µ mod Z. μ r+ J(E ) o / DE > a ! ~ u1 u0 A T B. Eichinger Killip-Simon problem on two disjoint intervals Functional models for Jacobi matrices Theorem (Jacobi theta function) There exists an analytic function θ1 on C with the following properties: θ1 (u ± 1) = −θ1 (u) θ1 (u ± τ ) = −e −iπτ e ∓2πiu θ1 (u) Moreover, the zeros of θ1 are given by v = m + nτ, m, n ∈ Z. Recall, that r+ is a meromorphic functions on RE with poles at {∞− , x0 } and zeros at {∞+ , x1 }. Thus, it is given by r+ (z, w ) = C θ1 (u − u∞+ )θ1 (u − u1 ) . θ1 (u − u∞− )θ1 (u − u0 ) B. Eichinger Killip-Simon problem on two disjoint intervals Functional models for Jacobi matrices Now we set B(u) = and k α (u) = C θ1 (u − u∞+ ) θ1 (u − u∞− ) θ1 (u − u0 ) , θ1 (u − uc− ) k α (u) K α (u) = p , k α (u∞+ ) where c− = 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∞+ ) e −i(u−u∞+ ) = . θ1 (u − u∞− + τ ) θ1 (u − u∞− ) e −i(u−u∞− ) Thus, we see that B(u + τ ) = e iµ B(u). B. Eichinger Killip-Simon problem on two disjoint intervals Functional models for Jacobi matrices Similarly, we can show that K α (u + τ ) = e iα K α (u), where α = u0 − uuc− . In general, we define K α−kµ (u) = θ1 (u − (u0 − kµ)) . θ1 (u − uc− ) Again, we see that K α−kµ (u + τ ) = e i(α−kµ) K α−kµ (u). B. Eichinger Killip-Simon problem on two disjoint intervals The space H 2 (α) Let Ω = C \ E . For α ∈ T, we define H 2 (α) as the space of multivalued, analytic functions f α (z) in Ω, which satisfy: |f α |2 has a harmonic majorant in Ω, g α (u) = f α (z), satisfies g α (u + τ ) = e iα g 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 Z α α hf1 , f2 i = f1α (x)f2α (x)ω(dx, ∞), E where ω(dx, ∞) denotes the harmonic measure of infinity of the domain Ω. It is given by i x −c p ω(dx, ∞) = . 2π s(z) B. Eichinger Killip-Simon problem on two disjoint intervals Functional models for Jacobi matrices Theorem Let enα (z) = B n (z)K α−kµ (z) Then en n ∈ N is a ONB of H 2 (α) en n ∈ Z is a ONB of L2 (α) Theorem Let enα be defined as above, then the multiplication operator z in L2 (α) with respect to the basis {enα } is a Jacobi matrix J = J(α): α α zenα = an (α)en−1 + bn (α)enα + an+1 (α)en+1 , B. Eichinger Killip-Simon problem on two disjoint intervals Sketch of the proof We only sketch, why the multiplication by z is a Jacobi matrix. We have r+ (z) = − θ1 (u − u∞+ )θ1 (u − u1 ) 1 . =C 2 (1) θ1 (u − u∞− )θ1 (u − u0 ) z − b0 + a1 r (z) That is, a0 r+ = −B K α−µ Kα (1) and a1 r+ = −B K α−2µ . K α−µ Thus we have zBK α−µ BK α−2µ a0 K α = (z − b ) − a ⇔ 0 1 BK α−µ K α−µ = a0 K α + b0 BK α−µ + a1 B 2 K α−2µ ⇔ ze1α = a0 e0α + b0 e1α + a1 e2α B. Eichinger Killip-Simon problem on two disjoint intervals Proof of the magic formula S −2 JS = J ⇔ 2µ ∈ Z. This is only a property of the spectrum not of J! Consider, in this case the basis {enα } has the following form {. . . , B −1 K α+µ , K α , BK α+µ , B 2 K α , . . . }. Recall µ = u∞+ − u∞− . There exists a function θ1 (u − u∞+ ) 2 ψ(z, w ) = = B(u)2 , θ1 (u − u∞− ) on RE . ψ + ψ −1 is a function on Ω = C \ E . Thus, ψ(z, w ) + 1 = T2 (z). ψ(z, w ) This is the magic formula. `2 L2 (α) en enα B. Eichinger S2 ψ J(α) z Killip-Simon problem on two disjoint intervals 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+ ) = B(u)Bc (u), θ1 (u − u∞− )θ1 (u − uc− ) where µ + µc ∈ Z. In this case we have ψ(z, w ) + 1 λ1 = λ 0 z + d0 + = ∆(z). ψ(z, w ) d1 − z B. Eichinger Killip-Simon problem on two disjoint intervals Functional models in the general case Let k α (z, c) = kcα (z) be the reproducing kernels of the space H 2 (α). We define Kψ (α) = H 2 (α) ψH 2 (α). Kψ is spanned by: f0α (z) = λ(α) kdα1 (z) kkdα1 (z)k and f1α (z) = Bd1 (z) k α+µ (z) , kk αµ k Thus, H 2 (α) = Kψ (α) ⊕ ψKψ (α) + ψ 2 Kψ (α) ⊕ . . . . Theorem The system of functions fnα ( ψ m f0α , n = 2m = ψ m f1α , n = 2m + 1 forms an orthonormal basis in H 2 (α) for n ∈ N and forms an orthonormal basis in L2 (α) for n ∈ Z. B. Eichinger Killip-Simon problem on two disjoint intervals Functional models in the general case, SMP-matrices Definition We call the operator A(α) ∈ L(`2 ), which corresponds to the multiplication by z in this basis a SMP-matrix. A(α) is two periodic. .. σ(A(α)) = E . . .. ... A A∗ B A A= A∗ B A∗ A .. . .. . B. Eichinger A= 0 0 , p0 p1 , p0 q0 + d0 p1 q0 . .. B = p1 q0 p1 q1 . Killip-Simon problem on two disjoint intervals By construction, we obtain the magic formula for SMP matrices. `2 L2 (α) en fnα S2 ψ A(α) z Theorem Let A(E ) be the set of all SMP matrices of period two with their spectrum on E. Then we have: A ∈ A(E ) ⇔ ∆(A) = λ0 A + d0 + λ1 (A − d1 )−1 = S 2 + S −2 B. Eichinger Killip-Simon problem on two disjoint intervals The Jacobi flow on SMP-matrices Definition We define the map F : A(E ) → J(E ), by F(A(α)) = J(α). Lemma Let kP+ A(α)e−1 k2 and {ak , bk } be the Jacobi parameters of J(α). Then we have p1 q1 = b−1 , p02 + q02 = a02 α =fα . It follows immediately, since e−1 −1 B. Eichinger Killip-Simon problem on two disjoint intervals The Jacobi flow on SMP-matrices Definition The Jacobi flow in A(E ) is defined by J (A(α)) = A(α − µ). A F /J J A B. Eichinger S F /J Killip-Simon problem on two disjoint intervals Explicit formulas for the Jacobi flow Theorem Let U(α) be the periodic 2 × 2-block diagonal unitary matrix given by (1) U(α) e2m e2m+1 = e2m e2m+1 u(α), where 1 u(α) = q p02 (α) + p12 (α) p0 (α) p1 (α) p1 (α) −p0 (α) (2) Then J A(α) := A(µ−1 α) = S −1 U(α)∗ A(α)U(α)S. B. Eichinger (3) Killip-Simon problem on two disjoint intervals