In a thin multidimensional layer we consider a second order differential PT-symmetric operator. The operator is of rather general form and its coefficients are arbitrary functions depending both on slow and fast variables. The PT-symmetry of the operator is ensured by the boundary conditions of Robin type with pure imaginary coefficient. In the work we determine the limiting operator, prove the uniform resolvent convergence of the perturbed operator to the limiting one, and derive the estimates for the rates of convergence. We establish the convergence of the spectrum of perturbed operator to that of the limiting one. For the perturbed eigenvalues converging to the limiting discrete ones we prove that they are real and construct their complete asymptotic expansions. We also obtain the complete asymptotic expansions for the associated eigenfunctions.

http://arxiv.org/abs/1403.4524

Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schroedinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schroedinger operators.

http://arxiv.org/abs/1102.5051
Spectral Theory (math.SP); Mathematical Physics (math-ph)