The Spectrum of Self-Adjoint Extensions associated with Exceptional Laguerre Differential Expressions
Abstract
Exceptional Laguerre-type differential expressions make up an infinite class of Schrödinger operators having rational potentials and one limit-circle endpoint. In this manuscript, the spectrum of all self-adjoint extensions for a general exceptional Laguerre-type differential expression is given in terms of the Darboux transformations which relate the expression to the classical Laguerre differential expression. The spectrum is extracted from an explicit Weyl $m$-function, up to a sign. The construction relies primarily on two tools: boundary triples, which parameterize the self-adjoint extensions and produce the Weyl $m$-functions, and manipulations of Maya diagrams and partitions, which classify the seed functions defining the relevant Darboux transforms. Several examples are presented.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2022
- DOI:
- arXiv:
- arXiv:2208.09459
- Bibcode:
- 2022arXiv220809459F
- Keywords:
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- Mathematics - Spectral Theory;
- Mathematics - Functional Analysis;
- 34L05;
- 33D45;
- 47E05;
- 47A10
- E-Print:
- 37 pages, 6 figures