N wells at a circle. Splitting of lower eigenvalues

Abstract

A stationary Schrödinger operator on R 2 with a potential V having N nondegenerate minima which divide a circle of radius r 0 into N equal parts is considered. Some sufficient asymptotic formulae for lower energy levels are obtained in a simple example. The ideology of our research is based on an abstract theorem connecting modes and quasi-modes of some self-adjoint operator A and some more detailed investigation of low energy levels in one well (in R d).

On the behaviour of the two-dimensional Hamiltonian $-\,{\rm{\Delta }}+\lambda [\delta (\vec{x}+{\vec{x}}_{0})+\delta (\vec{x}-{\vec{x}}_{0})]$ as the distance between the two centres vanishes

“…The possibility to solve this problem in that case was discussed in [18]. The case of N wells on the plane was studied in [25].…”