On a $\mathcal{PT}$-symmetric waveguide with a pair of small holes

A planar $\mathcal{PT}$-symmetric waveguide with a pair of small holes is considered. The waveguide is modeled by a planar infinite strip in which a pair of symmetric small holes is cut out. The operator is the Laplacian with $\mathcal{PT}$-symmetric boundary condition at the edges of the strip and Neumann condition at the boundaries of the holes. For this operator, the uniform resolvent convergence is established and the convergence rate is estimated. The effect of the generation by the holes of new eigenvalues from the boundary of the continuous spectrum is studied. Sufficient conditions for the existence and absence of such eigenvalues are obtained and the first terms of their asymptotic expansions are found.