Positive definite functions and spectral properties of the Schrödinger operator with point interactions
type: Item
instance of: scholarly article
title: Positive definite functions and spectral properties of the Schrödinger operator with point interactions
zbMATH Open document ID: 1244.47040
DOI: 10.1134/S0001434611070157
author: N. I. Goloshchapova
author: M. M. Malamud
author: V. P. Zastavny
published in: Mathematical Notes
publication date: 2012-03-13T00:00:00Z
review text: The principal object under consideration in the present paper is the differential expression \[ -\Delta+\sum_{j=1}^m \alpha_j\delta(\cdot -x_j), \qquad x_j\in\mathbb{R}^d, \quad \alpha_j\in\mathbb{R}, \quad m\in\mathbb{N}, \] and the corresponding minimal operator \(H_d\) in \(L^2(\mathbb{R})\) \[ H_d:=-\Delta, \qquad D(H_d):=\{f\in W_2^2(\mathbb{R}^d): f(x_j)=0, \;\;j=1,2,\dots,m\}, \] with deficiency indices \(n_{\pm}(H_d)=m\). The authors study the spectral properties of self-adjoint extensions of \(H_d\). As the first step, they obtain an explicit description of the domain of definition and boundary triples for \(H_d^*\) with \(d=2,3\). The main result of the paper states that the nonnegative spectrum of each self-adjoint extension \(H_{d,\theta}\) is absolutely continuous and has infinite multiplicity. The point spectrum \(\sigma_p (H_{d,\theta})\) consists of at most \(m\) negative eigenvalues, counting multiplicity, and for \(z\in\sigma_p (H_{d,\theta})\) the corresponding eigenfunction (for \(d=3\)) is of the form \[ \psi(x,z)=\sum_{j=1}^m c_j\,\frac{e^{i\sqrt{z}|x-x_j|}}{4\pi|x-x_j|}\,. \] The key ingredient is the positive-definiteness of certain matrices, which provides a link to the theory of positive definite functions in \(\mathbb{R}^d\).
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 6013935
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: radial positive definite functions
zbMATH Keywords: selfadjoint operators
zbMATH Keywords: Weyl function
zbMATH Keywords: spectrum
zbMATH Keywords: point interaction
zbMATH Keywords: deficiency index
zbMATH Keywords: Green's formula
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
cites work: Q3994511
cites work: Q5737276
cites work: Q3683061
cites work: Generalized resolvents and the boundary value problems for Hermitian operators with gaps
cites work: On the unitary equivalence of absolutely continuous parts of self-adjoint extensions
cites work: Q4059363
cites work: The von Neumann problem for nonnegative symmetric operators
full work available at URL: s0001434611070157
OpenAlex ID: W1992740860
description: scientific article; zbMATH DE number 6013935