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On positive Jacobi matrices with compact inverses

Pavel Šťovíček, Grzegorz Świderski

TL;DR

The paper develops a comprehensive operator-theoretic and spectral-analysis framework for positive Jacobi matrices with compact (or Schatten-class) inverses, linking spectral data to orthogonal polynomials via the three-term recurrence and Weyl/second-kind functions. It establishes concrete criteria for trace-class and Schatten-class inverses, derives the corresponding orthogonality measures, eigenvectors, and characteristic functions, and analyzes the asymptotic behavior of zeros and Christoffel-Darboux kernels under these spectral assumptions. A Birth-Death-process specialization yields explicit formulas and interlacing properties, including trace formulas for the inverse and Regulated determinants of truncated Jacobi matrices. The paper culminates with a construction of modified $q$-Laguerre polynomials, shown to be determinate and representable as Birth-Death polynomials, whose orthogonality measure is supported on roots of a $q$-Bessel function, revealing rich connections between orthogonal polynomials, special functions, and stochastic processes. These results collectively advance the understanding of spectral-discreteness phenomena, asymptotics of orthogonal polynomials, and explicit solvable models in the positive-Jacobi setting.

Abstract

We consider positive Jacobi matrices $J$ with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their zeros, the vague convergence of the zero counting measures and of the Christoffel--Darboux kernels on the diagonal. Particularly, if the inverse of $J$ belongs to some Schatten class, we identify the asymptotic behaviour of the sequence of orthogonal polynomials and express it in terms of its regularized characteristic function. In the even more special case when the inverse of $J$ belongs to the trace class we derive various formulas for the orthogonality measure, eigenvectors of $J$ as well as for the functions of the second kind and related objects. These general results are given a more explicit form in the case when $-J$ is a generator of a Birth--Death process. Among others we provide a formula for the trace of the inverse of $J$. We illustrate our results by introducing and studying a modification of $q$-Laguerre polynomials corresponding to a determinate moment problem.

On positive Jacobi matrices with compact inverses

TL;DR

The paper develops a comprehensive operator-theoretic and spectral-analysis framework for positive Jacobi matrices with compact (or Schatten-class) inverses, linking spectral data to orthogonal polynomials via the three-term recurrence and Weyl/second-kind functions. It establishes concrete criteria for trace-class and Schatten-class inverses, derives the corresponding orthogonality measures, eigenvectors, and characteristic functions, and analyzes the asymptotic behavior of zeros and Christoffel-Darboux kernels under these spectral assumptions. A Birth-Death-process specialization yields explicit formulas and interlacing properties, including trace formulas for the inverse and Regulated determinants of truncated Jacobi matrices. The paper culminates with a construction of modified -Laguerre polynomials, shown to be determinate and representable as Birth-Death polynomials, whose orthogonality measure is supported on roots of a -Bessel function, revealing rich connections between orthogonal polynomials, special functions, and stochastic processes. These results collectively advance the understanding of spectral-discreteness phenomena, asymptotics of orthogonal polynomials, and explicit solvable models in the positive-Jacobi setting.

Abstract

We consider positive Jacobi matrices with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their zeros, the vague convergence of the zero counting measures and of the Christoffel--Darboux kernels on the diagonal. Particularly, if the inverse of belongs to some Schatten class, we identify the asymptotic behaviour of the sequence of orthogonal polynomials and express it in terms of its regularized characteristic function. In the even more special case when the inverse of belongs to the trace class we derive various formulas for the orthogonality measure, eigenvectors of as well as for the functions of the second kind and related objects. These general results are given a more explicit form in the case when is a generator of a Birth--Death process. Among others we provide a formula for the trace of the inverse of . We illustrate our results by introducing and studying a modification of -Laguerre polynomials corresponding to a determinate moment problem.
Paper Structure (21 sections, 26 theorems, 271 equations)

This paper contains 21 sections, 26 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. More on positive Jacobi operators with a trace class inverse
  3. A criterion for the trace class
  4. Functions of the second kind, the orthogonality measure
  5. Jacobi matrices with trace resolvent related to generators of Birth-Death processes
  6. A criterion for the trace class in the case of generators of Birth-Death processes
  7. The characteristic function
  8. The function $\mathfrak{W}(z)$
  9. Some results for positive Jacobi matrices with compact inverses
  10. On the convergence of the zeros of orthogonal polynomials
  11. The zero counting measures
  12. The Christoffel--Darboux kernels
  13. The regularized characteristic function for Schatten classes
  14. Importance of the positivity
  15. The modified $q$-Laguerre polynomials
  16. ...and 6 more sections

Key Result

Theorem 2.1

Assume that and $J$ is positive-semidefinite. Then $J$ is self-adjoint and invertible. Moreover, $J^{-1}$ belongs to the trace class if and only if If so then $\mathop\mathrm{Tr}\nolimits(J^{-1})$ equals the sum in eq:sum-trace (including the sign).

Theorems & Definitions (63)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Remark
  • Theorem 2.3
  • proof
  • Remark
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • ...and 53 more