Uniform asymptotics for eigenvalues of model Schrödinger operator with small translation

Abstract

We consider a model Schrödinger operator with a constant coefficient on the unit segment and the Dirichlet and Neumann condition on opposite ends with a small translation in the free term. The value of the translation is small parameter, which can be both positive and negative. The main result is the spectral asymptotics for the eigenvalues and eigenfunctions with an estimate for the error term, which is uniform in the small parameter. For finitely many first eigenvalues and associated eigenfunctions we provide asymptotics in the small parameter. We prove that each eigenvalue is simple, and the system of eigenfunctions forms a basis in the space $L_2(0, 1).$