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Finite gap Jacobi matrices, I. The isospectral torus

Christiansen, Jacob S. LU ; Simon, Barry and Zinchenko, Maxim (2010) In Constructive Approximation 32(1). p.1-65
Abstract

Let e ⊂ ℝ be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.

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author
; and
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Covering map, Isospectral torus, Orthogonal polynomials
in
Constructive Approximation
volume
32
issue
1
pages
65 pages
publisher
Springer
external identifiers
  • scopus:77954088999
ISSN
0176-4276
DOI
10.1007/s00365-009-9057-z
language
English
LU publication?
no
id
ad747ba1-8727-493d-8618-138058cdf073
date added to LUP
2025-07-11 12:15:48
date last changed
2025-10-14 11:29:10
@article{ad747ba1-8727-493d-8618-138058cdf073,
  abstract     = {{<p>Let e ⊂ ℝ be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.</p>}},
  author       = {{Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim}},
  issn         = {{0176-4276}},
  keywords     = {{Covering map; Isospectral torus; Orthogonal polynomials}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{1--65}},
  publisher    = {{Springer}},
  series       = {{Constructive Approximation}},
  title        = {{Finite gap Jacobi matrices, I. The isospectral torus}},
  url          = {{http://dx.doi.org/10.1007/s00365-009-9057-z}},
  doi          = {{10.1007/s00365-009-9057-z}},
  volume       = {{32}},
  year         = {{2010}},
}