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Spectral properties of a q-Sturm-Liouville operator

Brown, B. Malcolm ; Christiansen, Jacob S. LU and Schmidt, Karl Michael (2009) In Communications in Mathematical Physics 287(1). p.259-274
Abstract

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.

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author
; and
publishing date
type
Contribution to journal
publication status
published
subject
in
Communications in Mathematical Physics
volume
287
issue
1
pages
16 pages
publisher
Springer
external identifiers
  • scopus:60449109148
ISSN
0010-3616
DOI
10.1007/s00220-008-0623-1
language
English
LU publication?
no
id
d33cccb7-d1eb-41bd-8085-76d0f045bbc5
date added to LUP
2025-07-11 12:17:30
date last changed
2025-10-14 13:29:38
@article{d33cccb7-d1eb-41bd-8085-76d0f045bbc5,
  abstract     = {{<p>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.</p>}},
  author       = {{Brown, B. Malcolm and Christiansen, Jacob S. and Schmidt, Karl Michael}},
  issn         = {{0010-3616}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{259--274}},
  publisher    = {{Springer}},
  series       = {{Communications in Mathematical Physics}},
  title        = {{Spectral properties of a q-Sturm-Liouville operator}},
  url          = {{http://dx.doi.org/10.1007/s00220-008-0623-1}},
  doi          = {{10.1007/s00220-008-0623-1}},
  volume       = {{287}},
  year         = {{2009}},
}