Homogeneous Schrödinger operators on half-line
type: Item
instance of: scholarly article
title: Homogeneous Schrödinger operators on half-line
zbMATH Open document ID: 1226.47049
author: Laurent Bruneau
author: Jan Dereziński
author: Vladimir Georgescu
published in: Annales Henri Poincaré
publication date: 2011-05-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.5569
review text: This article is devoted to a thorough study of the differential expression \(L_m:=-\partial_x^2+(m^2-1/4)x^{-2}\) on \(C^\infty_{c}(0,\infty)\), and of the operators it induces in the complex space \(L^2(0,\infty)\). These operators appear in the decomposition of the Aharonov-Bohm Hamiltonian. They are also of interest due to their role in the theory of special functions. The results state essentially that there is a unique holomorphic family \(\{H_m\}_{\operatorname{Re}\,m>-1}\) of closed operators in \(L^2\) such that \(H_m\) coincides with the closure of \(L_m\) for \(m\geq1\). The operators \(H_m\) are homogeneous of degree \(-2\) with respect to the group of dilations in \(L^2\). The spectrum and essential spectrum of \(H_m\) is \([0,\infty)\), independently of \(m\). The numerical range of \(H_m\) is calculated explicitly. If \(\operatorname{Re}\,m>-1\), \(\operatorname{Re}\,k>-1\) and \(\lambda\in\mathbb{C}\setminus[0,\infty)\), then \((H_m-\lambda)^{-1}- (H_k-\lambda)^{-1}\) is a compact operator. There are also results on the scattering theory for \(H_m\), e.g., an explicit expression for the wave operators. The proofs rest on an abstract study of operators that are homogeneous with respect to a strongly continuous group of unitary operators in a Hilbert space. Moreover, explicit formulas and estimates involving Bessel functions and the Hankel transform are employed.
reviewed by: Nils Ackermann
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 81Q12
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 5883583
zbMATH Keywords: homogeneous operator
zbMATH Keywords: holomorphic family of closed operators
zbMATH Keywords: Aharonov-Bohm Hamiltonian
zbMATH Keywords: Hankel transform
zbMATH Keywords: group of dilations
MaRDI profile type: Publication
OpenAlex ID: W1990114462
arXiv ID: 0911.5569
cites work: The Bessel differential equation and the Hankel transform
cites work: Q5526485
cites work: Weber–Schafheitlin-type integrals with exponent 1
cites work: The Hamiltonian (p2+m2)1/2−α/r near the critical value αc=2/π
cites work: SPECTRAL AND SCATTERING THEORY FOR THE AHARONOV–BOHM OPERATORS
cites work: Q4856171
cites work: Q3447431
DOI: 10.1007/S00023-011-0078-3
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description: scientific article