The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators

This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas.     

Spectral Properties of a <Emphasis Type="Italic">q</Emphasis>-Sturm–Liouville Operator

We study the spectral properties of a class of Sturm-Liouville type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, we construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. It is also shown that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively.     

On New Relations Between Spectral Properties of Jacobi Matrices and Their Coefficients

We study the spectral properties of Jacobi matrices. By using ``higher order' trace formulae we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. Complicated expressions for traces of some operators can be magically simplified allowing us to apply induction arguments. Our theorems are generalizations of a recent result of R. Killip and B. Simon [17].     

Gap-length mapping for periodic Jacobi matrices

We construct a real analytic isomorphism between periodic Jacobi operators and the spectral data formed by the gap lengths, the distances between the Dirichlet eigenvalues and the center of the corresponding gap, and some signs. This proves the uniqueness of the solution of the inverse problem and gives a characterization of the solution. Moreover, two-sided a priori estimates of periodic Jacobi operators in terms of gap lengths are obtained. Dedicated to the memory of B. M. Levitan     

Reflectionless Analytic Difference Operators II. Relations to Soliton Systems

Abstract

This is the second part of a series of papers dealing with an extensive class of analytic difference operators admitting reflectionless eigenfunctions. In the first part, the pertinent difference operators and their reflectionless eigenfunctions are constructed from given “spectral data”, in analogy with the IST for reflectionless Schrödinger and Jacobi operators. In the present paper, we introduce a suitable time dependence in the data, arriving at explicit solutions to a nonlocal evolution equation of Toda type, which may be viewed as an analog of the KdV and Toda lattice equations for the latter operators. As a corollary, we reobtain various known results concerning reflectionless Schrödinger and Jacobi operators. Exploiting a reparametrization in terms of relativistic Calogero–Moser systems, we also present a detailed study of N-soliton solutions to our nonlocal evolution equation.     

New Classes of Toda Soliton Solutions

We provide a detailed investigation of limits of N–soliton solutions of the Toda lattice as N tends to infinity. Our principal results yield new classes of Toda solutions including, in particular, new kinds of soliton–like (i.e., reflectionless) solutions. As a byproduct we solve an inverse spectral problem for one–dimensional Jacobi operators and explicitly construct tri–diagonal matrices that yield a purely absolutely continuous spectrum in (-1,1) and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of . Received: 16 October 1995/Accepted: 23 July 1996     

Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.     

Quantizer–dequantizer operators as a tool for formulating the quantization procedure

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.     

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parameterized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel–Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.     

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these models, have an interesting multiplicity structure. The results are obtained via a decomposition of the Laplacian into a direct sum of Jacobi matrices.     

Renormalized quantum tomography

Quantum tomography allows to expand operators over functions of a continuous set of spectral densities — the so-called quorum of observables. We present new nontrivial operator expansions for the quorum of quadratures of the harmonic oscillator, and introduce a general framework to generate new expansions based on the Kolmogorov construction.     

A generalized Talmi-Moshinsky transformation for few-body and direct interaction matrix elements

A set of basis states for use in evaluating matrix elements of few-body system operators is suggested. These basis states are products of harmonic oscillator wave functions having as arguments a set of Jacobi coordinates for the system. We show that these harmonic oscillator functions can be chosen in a manner that allows such a product to be expanded as a finite sum of the corresponding products for any other set of Jacobi coordinates. This result is a generalization of the Talmi-Moshinsky transformation for two equal-mass particles to a system of any number of particles of arbitrary masses. With the help of our method the multidimensional integral which must be performed to evaluate a few-body matrix element can be transformed into a sum of products of three dimensional integrals. The coefficients in such an expansion are generalized Talmi-Moshinsky coefficients. The method is tested by calculation of a matrix element for knockout scattering for a simple three-body system. The results indicate that the method is a viable calculational tool.     

Formulas for q-Spherical Functions Using Inverse Scattering Theory of Reflectionless Jacobi Operators

We study the spectral problem associated to a Ruijsenaars-type (q-)difference version of the one-dimensional Schrödinger operator with Pöschl-Teller potential. The eigenfunctions are constructed explicitly with the aid of the inverse scattering theory of reflectionless Jacobi operators. As a result, we arrive at combinatorial formulas for basic hypergeometric deformations of zonal spherical functions on odd-dimensional hyperboloids and spheres.     

Feynman quantization: An operator approach to path integration over commuting and anticommuting variables

We present an operator quantization scheme on a continuous direct product of Hilbert spaces over a time interval as an extension of the quantization using Feynman path integrals. We define the continuous direct product as a Hilbert space with two principal bases: the Fock and the Feynman ones. The Fock basis, defined by a complete set of commuting operators at different times, serves for a definition of the operator calculus. The Feynman basis, simultaneously diagonalizing the complete set of commuting operators, leads to path integrals constructed without time slicing as a spectral representation of certain operator functions. The construction of quantum theory and the corresponding path integrals for the harmonic oscillator is demonstrated both in the configuration and phase spaces. The extension of the theory to coherent states and anticommuting variables is performed.     

Probabilistic Averages of Jacobi Operators

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ^ρ (mod 1)) for ρ > 0 not an integer.     

Lieb–Thirring inequalities for complex finite gap Jacobi matrices

We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap almost periodic Jacobi matrices.     

Rotation and vibration of diatomic molecule oscillator with hyperbolic potential function

The explicit expressions of energy eigenvalues and eigenfunctions of bound states for a three-dimensional diatomic molecule oscillator with a hyperbolic potential function are obtained approximately by means of the hypergeometric series method. Then for a one-dimensional system, the rigorous solutions of bound states are solved with a similar method. The eigenfunctions of a one-dimensional diatomic molecule oscillator, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.     

A generalized spectral duality theorem

We establish a version of the spectral duality theorem relating the point spectrum of a family of^*-representations of a certain covariance algebra to the continuous spectrum of an associated family of^*-representations. Using that version, we prove that almost all the images of any element of a certain space of fixed points of some^*-automorphism of an irrational rotation algebra via standard^*-representations of the algebra inl ² do not have pure point spectrum over any non-empty open subset of the common spectrum of those images. As another application of the spectral duality theorem, we prove that if almost all the Bloch operators associated with a real almost periodic function on have pure point spectrum over a Borel subset of , then almost all the Schrödinger operators with potentials belonging to the compact hull of the translates of this function have, over the same set, purely continuous spectrum.Dedicated to Professor Marek Burnat     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.