Spectral theory of the Klein-Gordon equation in Pontryagin spaces
type: Item
instance of: scholarly article
title: Spectral theory of the Klein-Gordon equation in Pontryagin spaces
zbMATH Open document ID: 1114.47038
author: B. Najman
author: Christiane Tretter
author: Heinz Langer
published in: Communications in Mathematical Physics
publication date: 2007-02-13T00:00:00Z
review text: The authors consider the abstract Klein-Gordon equation \[ ((d/dt- iV)^2+ H_0)u= 0, \] where \(H_0\) is a strictly positive self-adjoint operator in a Hilbert space \(H\) with scalar product \((\cdot,\cdot)\) and \(V\) is a symmetric operator in \(H\) such that \(D(H^{1/2}_0)\subset S(V)\), \(S:= VH^{-1/2}_0= S_0+ S_1\) with \(\| S_0\|< 1\) and \(S_1\) compact and \(1\in \rho(S^* S)\). This equation can be transformed into a first-order differential equation, which leads to the operator \[ A:= \begin{pmatrix} 0 & I\\ H & 2V\end{pmatrix},\;H:= H^{1/2}_0(I- S^*S) H^{1/2}_0,\;D(A):= D(H)\oplus D(H^{1/2}_0). \] \(A\) is a self-adjoint operator in the space \(K:= D(H^{1/2}_0)\oplus H\) equipped with the indefinite inner product \(\langle X,X'\rangle:= ((I- S^* S)H^{1/2}_0 x, H^{1/2}_0 x')+ (y,y')\), \(X= (xy)^t\), \(X'= (x'y')^t\in K\). The results of the paper are the following: (1) \(K\) is a Pontryagin space with finite negative integer \(\kappa\), where \(\kappa\) is the number of negative eigenvalues of the operator \(I- S^*S\). (2) \(A\) has a spectral function with at most finitely many critical points. (3) The non-real spectrum of \(A\) is symmetric with respect to the real axis and consists of at most \(\kappa\) pairs of eigenvalues \(\lambda,\overline\lambda\) of finite type. (4) \(\sigma_{\text{ess}}(A)\) is real and \(\sigma_{\text{ess}}\cap(-\alpha, \alpha)= \phi\), where \(\alpha:= (1-\| S_0\|)m\), \(H_0\geq m^2\). (5) \(A\) generates a strongly continuous group of unitary operators in \(K\), which is uniformly bounded. The conditions on the potential \(V\) are illustrated for the Klein-Gordon equation in \(\mathbb{R}^n\); they include potentials consisting of a Coulomb part and an \(L^p\)-part with \(n\leq p<\infty\).
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 5125447
zbMATH Keywords: Klein-Gordon equation
zbMATH Keywords: Pontryagin space
zbMATH Keywords: energy inner product
zbMATH Keywords: spectral properties
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
full work available at URL: s00220-006-0022-4
OpenAlex ID: W1982919660
cites work: Q4692935
cites work: Superradiance and scattering of the charged Klein-Gordon field by a step-like electrostatic potential
cites work: Remarks on the Schrödinger operator with singular complex potentials
cites work: On the existence of wave operators for the Klein-Gordon equation
cites work: Scattering theory for the Klein-Gordon equation
cites work: Q3764747
cites work: Classes of linear operators. Vol. I
cites work: Q2752819
cites work: Q3866678
cites work: Q4143803
cites work: Q3955947
cites work: Relativistic quantum theory for charged spinless particles in external vector fields
cites work: Spectral and scattering theory for the Klein-Gordon equation
cites work: Q4186887
cites work: Q3879799
cites work: Localization of the critical points of Klein-Gordon type operators
cites work: Eigenvalues of the Klein-Gordon equation
cites work: Q4066072
cites work: Q4185363
cites work: The Klein-Gordon equation and scattering theory
cites work: Q5640289
cites work: Q4003552
cites work: On the nonrelativistic limit of the bound states of the Klein-Gordon equation
DOI: 10.1007/S00220-006-0022-4
description: scientific article; zbMATH DE number 5125447