Singular point-like perturbations of the Bessel operator in a Pontryagin space
This paper is elaborated in an (introduction plus) 6 sections, namely: 1. Pontryagin space realization of singular perturbations of a nonnegative operator. 2. A model for singular perturbations of a nonnegative operator. 3. Point-like perturbation of the Bessel operator. 4. Spectral properties of \(H^t\). 5. Eigenfunction expansions. 6. Scattering matrices. To produce a general but very concise presentation of the content, it's useful to cite the authors' abstract: ``The spectral problem for the Bessel equation of order \(\nu\) on \((0,\infty)\) in the case \(0< \nu<1\) is closely related to the Nevanlinna functions \[ Q(z)= -\pi(- z)^\nu/(2\sin\pi\nu). \] If \(\nu> 1\) and \(\nu\neq 2,3,\dots\), this function belongs to the generalized Nevanlinna class \(N_m\), \(m= [{v+1\over 2}]\). A natural question appears: To what spectral problem does this function correspond? We answer this and related questions using Pontryagin space operator realizations of suitable singular point-like perturbations of the Bessel operator. In this paper we discuss the spectra of these realizations and we derive eigenfunction expansions via related wave operators''.
scientific article; zbMATH DE number 1496034
scientific article