turkmath.org

The main object of my talk is the Schr\"{o}dinger operator $L(V)$ with a matrix potential on a $d-$ dimensional rectangle with Neumann boundary conditions. The operator $L(V)$ is considered as a perturbation of the free Hamiltonian $L(0)$, when $V=0$. Perturbative analysis of eigenvalues and eigenfunctions of $L(V)$ meets a serious difficulty due to presence of the very close eigenvalues of unperturbed operator. In general, this leads to the so-called `small denominators' problem. However, the set of unperturbed eigenvalues can be splitted into two parts called resonance and non-resonance domains, respectively. This opens a door to study the perturbations of each group for obtaining various asymptotic formulas for the eigenvalues and eigenfunctions of $L(V)$. I will talk about this problem which is a part of my PhD dissertation.