Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.

Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.

Background: The discovery of solitary waves of translation goes back to Scott Russell in 1834, and during the remaining part of the 19th century the true nature of these waves remained controversial. It was only with the derivation by Korteweg and de Vries in 1895 of what is now called the Korteweg-de Vries (KdV) equation, that the one-soliton solution and hence the concept of solitary waves was put on a firm basis. 1 An extraordinary series of events took place around 1965 when Kruskal and Zabusky, while analyzing the numerical results of Fermi, Pasta, and Ulam on heat conductivity in solids, discovered that pulselike solitary wave solutions of the KdV equation, for which the name "solitons" was coined, interact elastically. This was followed by the 1967 discovery of Gardner, Greene, Kruskal, and Miura that the inverse scattering method allows one to solve initial value problems for the KdV equation with sufficiently fast-decaying initial data. Soon thereafter, in 1968, Lax found a new explanation of the isospectral nature of KdV solutions using the concept of Lax pairs and introduced a whole hierarchy of KdV equations. Subsequently, in the early 1970s, Zakharov and Shabat (ZS), and Ablowitz, Kaup, Newell, and Segur (AKNS) extended the inverse scattering method to a wide class of nonlinear partial differential equations of relevance in various scientific contexts ranging from nonlinear optics to condensed matter physics and elementary particle physics. In particular, solitons found numerous applications in classical and quantum field theory and in connection with optical communication devices. Another decisive step forward in the development of completely integrable soliton equations was taken around 1974. Prior to that period, inverse spectral 1 With hindsight, though, it is now clear that other researchers, such as Boussinesq, derived the KdV equation and its one-soliton solution prior to 1895, as described in the notes to Section 1.1.

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 Schrödinger 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.

This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian − C ∞ 0 ( ) in L 2 ( ; d n x). Here, the domain belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C 1,r , for r > 1/2. Second, we establish Kreȋn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on (a, ∞) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is selfadjoint (equivalently, the associated minimally defined Schrödinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators.

Abstract. We study inverse spectral analysis for finite and semi-infinite Jacobi matrices H. Our results include a new proof of the central result of the inverse theory (that the spectral measure determines H). We prove an extension of Hochstadt's theorem (who proved the result in the case n = N ) that n eigenvalues of an N × N Jacobi matrix, H, can replace the first n matrix elements in determining H uniquely. We completely solve the inverse problemThere is an enormous literature on inverse spectral problems for − [1,29,[56][57][58][59][60]64] and references therein), but considerably less for their discrete analog, the infinite and semi-infinite Jacobi matrices (see, e.g., [3, 4, 6-8, 13-22, 24, 26-28, 30, 32, 37, 38, 42-44, 50-52, 61-63, 66, 67, 69-71]) and even less for finite Jacobi matrices (where references include, e.g., [9,10,23,25,39,40,41,[45][46][47][48]). Our goal in this paper is to study the last two problems using one of the most powerful tools from the spectral theory of − d 2 dx 2 + V (x), the m-functions of Weyl. Explicitly, we will study finite N × N matrices of the form:

Abstract. 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 Schrödinger 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.