Krein's formula and one-dimensional multiple-well

AbstractIt is shown that all the discrete eigenvalues of a one-dimensional Schrödinger operator with a multiple well potential possess an asymptotic expansions in power of h̵12 when h̵ → 0. A formula for all the coefficients of these expansions is given. The method uses two main tools: Perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article. As a by-product of this work, the exponential localization of eigenvectors when h̵ goes to zero can be proved