Spectral properties of the generalized Friedrichs model
type: Item
instance of: scholarly article
title: Spectral properties of the generalized Friedrichs model
zbMATH Open document ID: 1208.47007
author: E. R. Akchurin
published in: Theoretical and Mathematical Physics
publication date: 2010-09-24T00:00:00Z
review text: The author studies the operator \(A\) acting in the Hilbert space \[ \mathbb{C}\oplus L_2(M,d\mu) \] given by \[ (AF)_0=e_0f_0+\alpha\int_M\bar{v}(q)f_1(q)\,d\mu(q), \] \[ (AF)_1(q)=\alpha v(q)f_0+a(q)f_1(q)+\alpha\int_M D(q,q')f_1(q') \,d\mu(q'), \] where \(F=(f_0,f_1)\). This operator is called a generalized Friedrichs operator. The main results of this paper extend to this class of operators results previously known for the class of Friedrichs operators. The results are: {\parindent=5mm \begin{itemize}\item[1.] For sufficiently small values of \(\alpha\), the spectrum of \(A\) consists of an absolutely continuous part equal to the range of the function \(a\), and a possible single eigenvalue, smaller than the minimum of \(a\). \item[2.] The existence and completeness of wave operators related to \(A\). \end{itemize}} Suitable conditions on \(a\), \(v\), \(D\) and the constant \(\alpha>0\) are imposed to this effect. The author mentions that similar results (with similar conditions) where stated in [\textit{N.\,Angelescu, R.\,A.\thinspace Minlos} and \textit{V.\,A.\thinspace Zagrebnov}, Rev.\ Math.\ Phys.\ 17, No.\,10, 1111--1142 (2005; Zbl 1086.81040)], though complete proofs are given here.
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 5790865
zbMATH Keywords: generalised Friedrichs model
zbMATH Keywords: resolvent
zbMATH Keywords: spectrum
zbMATH Keywords: generalised eigenvector
zbMATH Keywords: wave operator
reviewed by: Esteban Andruchow
MaRDI profile type: Publication
full work available at URL: s11232-010-0032-4
OpenAlex ID: W2041376113
cites work: Über die Spektralzerlegung eines Integraloperators
cites work: Some special properties of the generalized Friedrichs model
cites work: Q5517170
cites work: On the perturbation of continuous spectra
cites work: Q3996935
cites work: Q4273956
cites work: LOWER SPECTRAL BRANCHES OF A PARTICLE COUPLED TO A BOSE FIELD
cites work: Random walks in a random (fluctuating) environment
cites work: The spectrum of two-particle bound states for the transfer matrices of Gibbs fields (an isolated bound state)
cites work: Q5562285
cites work: Q5332803
cites work: Q3920175
cites work: Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive
cites work: Q4185363
cites work: Q4191141
cites work: Q5571194
DOI: 10.1007/S11232-010-0032-4
Recommended article: Q4249047
Recommended article: ON A CLASS OF GENERALIZED FREDHOLM OPERATORS, VII
Recommended article: Q3737794
Recommended article: Expansions in generalized eigenfunctions of selfadjoint operators
Recommended article: Q4891819
Recommended article: ON A CLASS OF GENERALIZED FREDHOLM OPERATORS, V
Recommended article: New branches of the essential spectrum of a 2 × 2 operator matrix
Recommended article: Q4796956
Recommended article: Q4955948
Recommended article: Q3736306
description: scientific article; zbMATH DE number 5790865