Fredholm property and essential spectrum of pseudodifferential operators with operator-valued symbols
The notion of a limit operator was introduced into the theory of pseudodifferential operators \((\psi\text{do's})\) by \textit{B.\,V.\thinspace Lange} and \textit{V.\,S.\thinspace Rabinovich} in [Math.\ USSR, Sb.\ 57, 183--194 (1987); translation from Mat.\ Sb., Nov.\ Ser.\ 129(171), No.~2, 175--185 (1986; Zbl 0611.47039)] and, for a certain class of symbols, they showed that the corresponding \(\psi\text{do's}\) have the Fredholm property (or, similarly, have a purely discrete spectrum) if and only if the limit operators are invertible (or have a purely discrete spectrum). The authors of the paper under review consider \(\psi\text{do's}\) with operator-valued symbols which have the property that their essential spectra are the union of the spectra of their limit operators. As an example, they consider a selfadjoint Schrödinger operator \(S\) with a smooth and bounded potential on a cylindrical domain \(\mathbb{R}^n\times\Omega\) (with \(\Omega\subset\mathbb{R}^m\) a bounded domain with a smooth boundary on which a Robin boundary condition is prescribed). In this case, the limit operators have a purely discrete spectrum and the essential spectrum of \(S\) is a half-line, the bottom of which is given by the infimum of their spectra.
scientific article; zbMATH DE number 5858850
scientific article