The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip

The Schrödinger operator in a thin infinite strip with $\mathcal{PT}$-symmetric boundary conditions and a localized potential is studied. The case of a virtual level on the threshold of the essential spectrum of an efficient one-dimensional operator is considered. Sufficient conditions for the transformation of this level into an isolated eigenvalue are obtained and the first terms of the asymptotic expansion are calculated for this eigenvalue. Sufficient conditions for the absence of such an eigenvalue are also obtained.