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

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.
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