Eigenvalues of the reference operator and semiclassical resonances

We prove that the estimate of the number of the eigenvalues in intervals , of the reference operator L^#(h) related to a self-adjoint operator L(h) is equivalent to the estimate of the integral over [λ−δ,λ+δ] of the sum of harmonic measures associated to the resonances of L(h) lying in a complex neighborhood Ω of λ>0 and the number of the positive eigenvalues of L(h) in [λ−δ,λ+δ]. We apply this result to obtain a Breit-Wigner approximation of the derivative of the spectral shift function near critical energy levels.     

The local index formula in semifinite Von Neumann algebras I: Spectral flow

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is ‘almost’ a (b,B)-cocycle in the cyclic cohomology of A.     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

Equivalence,linearization, and decomposition of holomorphic operator functions

This paper discusses the problem of classifying holomorphic operator functions up to equivalence. A survey is given in 41 of the main results about equivalence classes of holomorphic matrix functions and holomorphic Fredholm-operator functions. In 42, it is shown that given a holomorphic function A on a bounded domain Ω into a space of bounded linear operators between two Banach spaces, it is possible to extend the operators A(λ) (for each λ ? Ω) by an identity operator I_Z in such a way that the extended operator function A(·) ⊕ I_Z is equivalent on Ω to a linear function of λ, T ? λI. Other versions of this “linearization by extension” are described, including the cases of entire functions and polynomials (where Ω = $\text{C}$). As an application of these results, we consider the operator function equation $\text{A}{}_2\text{(λ) Z}{}_2\text{(λ) + Z}{}_1\text{(λ) A}{}_1\text{(λ) = C(λ), λ ? Ω}$, (1) and explicitly construct the solutions Z₁ and Z₂. The formulas for Z₁ and Z₂ seem to be new, even when A₁, A₂ and C are matrix polynomials. The existence of solutions of (¹) makes it possible to analyze an operator function A whose spectrum decomposes into pairwise disjoint compact subsets σ₁, …, σ_n of Ω. In this case, a suitable extension of A is equivalent on Ω to a direct sum of operator functions, A₁, …, A_n, such that the spectrum of A_i is σ_i (i = 1, …, n). In the final section of the paper, we discuss the relation between local and global equivalence on Ω, and show that there exist operator functions A and B which are locally equivalent on Ω, but admit no extensions (of the sort considered in this paper) which are globally equivalent on Ω.     

Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

We study the spectral shift function s(λ,h) and the resonances of the operator P(h)=-Δ+V(x)+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s(λ,h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s(λ,h). We establish an upper bound O(h^-n+1) for the number of the resonances of P(h) lying in a disk of radius h.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Toeplitz matrices and determinants with Fisher-Hartwig symbols

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ(A) and σ(B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ(A). If A is any self-adjoint operator, there is an associated function κ_A : $\text{N}$ ∪ {∞} → ($\text{N}$ ∪ {0, ∞}) × {0,1} defined in this paper. If $\text{F}$ denotes the collection of all functions from $\text{N}$ ∪ {∞} into ($\text{N}$ ∪ {0,∞}) × {0,1}, then $\text{F}$ is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ_A = κ_B. The paper ends with some comments on the corresponding problem for normal operators.     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

Lipschitz lower sigma-exponents of linear differential systems

For the lower sigma-exponent of the linear differential system ? = A(t)x, x ∈ R ⁿ , t ≥ 0, defined by the formula Δ_σ(A) ≡ inf_λ[Q]≤-σ λ ₁(A + Q), σ > 0, on the basis of the lower characteristic exponents λ ₁(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ ?σ < 0 of perturbations Q, we prove the following general form as a function of the parameter σ > 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ ₀,+∞), σ ₀ ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ ₀], we prove the existence of a linear system with coefficient matrix A _f (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δ_σ(A _f ) coincides with the function f(σ) on the entire interval (0,+∞).     

Uniform,on the entire axis,convergence of the spectral expansion for Schrödinger operator with a potential-distribution

A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W ₂ ^?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.     

Rank one perturbations at infinite coupling in Pontryagin spaces

In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)⁻¹ under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H^∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H^∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H^∞.     

Mourre's inequality and embedded bound states

A self-adjoint operator H with an eigenvalue λ embedded in the continuum spectrum is considered. Boundedness of all operators of the form AnP is proved, where P is the eigenprojection associated with λ and A is any self-adjoint operator satisfying Mourre's inequality in a neighborhood of λ and such that the higher commutators of H with A up to order n+2 are relatively bounded with respect to H.     

Multiple operator integrals and higher operator derivatives

In this paper we consider the problem of the existence of higher derivatives of the function t??(A+tK), where ? is a function on the real line, A is a self-adjoint operator, and K is a bounded self-adjoint operator. We improve earlier results by Sten’kin. In order to do this, we give a new approach to multiple operator integrals. This approach improves the earlier approach given by Sten’kin. We also consider a similar problem for unitary operators.     

Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ₀ and is boundedly invertible in the endpoints of Δ₀, then the ‘embedding’ of the original Hilbert space into the Hilbert space , where the linearization of A(z) acts, is in fact an isomorphism between a subspace of and . As a consequence, properties of the local spectral function of A(z) on Δ₀ and a so-called inner linearization of the operator function A(z) in the subspace are established.      

An abstract Szeg? theorem

Given a semi-group U(t) of bounded linear operators with bounded self-adjoint generator A we estimate the logarithm of the section determinants of U(t) in terms of A. When A is subject to an additional condition, which is related to so-called Følner sequences of orthogonal projections, this estimate implies a Szeg? type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when A is a Toeplitz operator or a compact operator.     

Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

The matrix-valued Weyl-Titchmarsh functions M(λ) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N×N Weyl-Titchmarsh functions) corresponding to N×N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.     

Operator Hölder-Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q−Qf(A) and quasicommutators f(A)Q−Qf(B). Finally, we estimate the norms of finite differences for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.     

A graph which embeds all small graphs on any large set of vertices

For certain cardinals λ and κ a colouring P:[λ]²→λ is constructed such that if X ϵ[λ]^λ and Q:[κ]²→λ, then there is a one-to-one function i:κ→X such that P(i″A)=Q(A) for every Aϵ[κ]². Additional results are obtained.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)