On the essential spectrum of the relativistic magnetic Schrödinger operator.
type: Item
instance of: scholarly article
title: On the essential spectrum of the relativistic magnetic Schrödinger operator.
zbMATH Open document ID: 1147.81304
author: Yanyan Li
published in: Osaka Journal of Mathematics
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1153492940
review text: Introduction: In a series of papers one has studied the Weyl quantized Hamiltonian of a relativistic spinless particle with a magnetic vector potential : \[ [Op^w (\lambda_a)u](x) = (2\pi)^{-n}\int\int e^{i\langle x-y,\xi\rangle} \lambda_a\biggl(\frac{x+y}{2},\xi\biggr)u(y)\,dy\,d\xi, \] where \(\lambda_a(x,\xi) = (|\xi -a(x)|^2 + 1)^{1/2}.\) For simplicity we suppose here that the mass of the particle is equal to 1. All the differential and pseudodifferential operators considered in this paper are, possibly unbounded, operators in \(L^2(\mathbb R^n)\), defined on the Schwartz space \(\mathcal S\) of rapidly decreasing smooth functions on \(\mathbb R^n\). In [\textit{M. Nagase} and \textit{T. Umeda}, J. Funct. Anal. 92, No. 1, 136--154 (1990; Zbl 0715.46051)] it was proved that if the derivatives of any positive order of \(a\in C^\infty(\mathbb R^n;\mathbb R^n)\) are bounded, then \(Op^w (\lambda_a)\) is essentially selfadjoint on \(\mathcal S\). Let \(h_a\) be its unique selfadjoint extension. In [Osaka J. Math. 30, No. 4, 839--853 (1993; Zbl 0814.35138)] \textit{M. Nagase} and \textit{T. Umeda} proved that if \(a\) itself is bounded and if all its derivatives converge to zero at infinity, then the essential spectrum of \(h_a\) is equal to the essential spectrum of \(\sqrt{H_a+1}\), where \(H_a\) is the quantum nonrelativistic magnetic Hamiltonian with vector potential, i.e. the selfadjoint operator generated by the differential operator \((D-a(X))^2\). We prove in this paper that the essential spectra of \(h_a\) and of \(\sqrt{H_a+1}\) are still equal if we drop the condition of boundedness of \(a\). Thus, vector potentials which behave at infinity as \(|x|^{1-\varepsilon}\), \(\varepsilon\) positive and arbitrary small, are allowed. More precisely, the main result of the paper is the following. Theorem 1.1. Suppose that: (i) the vector potential \(a\in C^\infty(\mathbb R^n;\mathbb R^n)\) is such that \[ \lim_{|x|\to\infty} \partial\beta a(x) = 0\qquad \forall \be…
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 1874594
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1874594