Fritz Gesztesy

On Matrix–Valued Herglotz Functions

We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued Schrodinger and Dirac–type operators. Special emphasis is devoted to appropriate matrix–valued extensions of the well–known Aronszajn–Donoghue theory concerning support properties of measures in their Nevanlinna–Riesz–Herglotz representation. In particular, we study a class of linear fractional transformations MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(z) is invariant under these linear fractional transformations. Additional applications discussed in detail include self–adjoint finite–rank perturbations of self–adjoint operators, self–adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrodinger operator H = -(d^(2)/(dx^(2)) + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.

A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

We continue the study of the A-amplitude associated to a half-line Schrodinger operator, - d^2/dx^2 + q in L^2((0,b)), b ≤ ∞ A is related to the Weyl-Titchmarsh m-function via m(-k^2) = -k- ʃ^a_0 A(α)e^(-2αk) dα+O(e^(-(2α-Є)k)) for all Є > 0. We discuss five issues here. First, we extend the theory to general q in L^1((0,α)) for all a, including qs which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure p: A(α) = -2 ^ʃ∞_(-∞)λ^(-1/2) sin (2α√λ) dp(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b < ∞. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.

Weyl–Titchmarsh -Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies

Introduction The Toda hierarchy, recursion relations, and hyperelliptic curves The stationary Baker-Akhiezer function Spectral theory for finite-gap Jacobi operators Quasi-periodic finite-gap solutions of the stationary Toda hierarchy Quasi-periodic finite-gap solutions of the Toda hierarchy and the time-dependent Baker-Akhiezer function The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy Spectral theory for finite-gap Dirac-type difference operators Quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy Hyperelliptic curves of the Toda-type and theta functions Periodic Jacobi operators Examples,

Algebro-Geometric Solutions of the Camassa-Holm hierarchy

g-0,1

M-FUNCTIONS AND INVERSE SPECTRAL ANALYSIS FOR FINITE AND SEMI-INFINITE JACOBI MATRICES

Acknowledgments Bibliography.

Picard potentials and Hill's equation on a torus

We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy and describe the associated isospectral torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we rely on Weyl-Titchmarsh theory for singular (canonical) Hamiltonian systems. We also briefly discuss real-valued algebro-geometric solutions with a cusp behaviour. While we focus primarily on the case of stationary algebro-geometric CH solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.

The xi function

We study inverse spectral analysis for finite and semi-infinite Jacobi matricesH. Our results include a new proof of the central result of the inverse theory (that the spectral measure determinesH). We prove an extension of the theorem of Hochstadt (who proved the result in casen = N) thatn eigenvalues of anN × N Jacobi matrixH can replace the firstn matrix elements in determiningH uniquely. We completely solve the inverse problem for (δn, (H-z)-1 δn) in the caseN < ∞.

An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

Hills equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness. Of particular importance in the last 20 years is its connection with the KdV hierarchy and hence with integrable systems. We show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials, a major open problem in the field. Specifically, we show that elliptic finitegap potentials of Hills equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable, complementing a classical theorem of Picard. The intimate connection between Picards theorem and elliptic finite-gap solutions of completely integrable systems is established in this paper for the first time. In addition, we construct the hyperelliptic Riemann surface associated with a finitegap potential (not necessarily elliptic), i.e., determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with Hills equation. These multiplicities are intimately correlated with the pole structure of the diagonal Greens function of the operator H = d2/dx2+q(x) in L2(R). Our construction is new in the present general complex-valued periodic finite-gap case. Before describing our approach in some detail, we shall give a brief account of the history of the problem involved. This theme dates back to a 1940 paper of Ince [43] who studied what is presently called the Lam6-Ince potential