Eigenvalue problems on exterior domains and Dirichlet to Neumann maps (Q596253)

The author considers the eigenvalue problem for the Schrödinger equation with a complex valued potential \(q\) in an exterior domain subject to third kind boundary conditions. It is assumed that \(q\) has a limit \(q(\infty)\) at infinity. The problem is perturbed in a first step by cutting off \(q\) outside of a large ball to obtain a compactly supported potential \(Q\) but still keeping the infinite domain. In a second step the obtained problem is further reduced by considering it in a finite domain subject to the condition that it contains the support of \(Q\) and imposing Dirichlet conditions on the outer boundary. For both kinds of perturbation point spectral inclusion and spectral exactness are proved. A main ingredient in the proofs is the analysis of approximate Dirichlet to Neumann maps on the inner boundary and the detailed knowledge of the solution of the Schrödinger equation with \(q=0\) in an annular domain. In a number of numerical examples, mostly for the one-dimensional case, the possibility of spurious eigenvalues is discussed although by the theoretical results they should not exist.