D. I. Borisov, M. Znojil, “On eigenvalues of a <nobr>$\mathscr{P\!T}$</nobr>-symmetric operator in a thin layer”, Mat. Sb., 2017, Volume 208, Number 2,Pages <nobr>3

This article is cited in 9 papers

On eigenvalues of a $\mathscr{P\!T}$-symmetric operator in a thin layer

D. I. Borisov^abc, M. Znojil^d

^a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa
^b Bashkir State Pedagogical University, Ufa
^c University of Hradec Králové, Czech Republic
^d Nuclear Physics Institute of the Czech Academy of Sciences, Řež, Czech Republic

Abstract: We consider an elliptic operator with variable coefficients in a thin three-dimensional layer with $\mathscr{P\!T}$-symmetric boundary conditions. We study the effect of the appearance of isolated eigenvalues at the edges of the gaps in the essential spectrum. We obtain sufficient conditions that guarantee that such eigenvalues either exist or are absent near a given edge of a gap. In the case of existence, the first terms in the asymptotic expansion of these emerging eigenvalues are calculated.
Bibliography: 34 titles.

Keywords: thin domain, $\mathscr{P\!T}$-symmetric operator, edge of a gap, asymptotics, periodic operator.

UDC: 517.956+517.958

MSC: Primary 35P15; Secondary 35P20, 47F05

Received: 05.01.2016 and 21.05.2016

DOI: 10.4213/sm8657