The Spectrum of Self-Adjoint Extensions associated with Exceptional Laguerre Differential Expressions

TL;DR

This paper characterizes the spectra of all self-adjoint extensions of exceptional Laguerre differential operators, linking them to classical Laguerre operators via Darboux transformations and explicit Weyl m-functions.

Contribution

It provides a comprehensive spectral analysis of exceptional Laguerre operators using boundary triples and Darboux transformations, extending classical results to a broader class.

Findings

Spectra are explicitly described via Weyl m-functions.

Self-adjoint extensions are parameterized by boundary triples.

Examples illustrate the spectral classification process.

Abstract

Exceptional Laguerre-type differential expressions make up an infinite class of Schr\"odinger 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.