Researcher's Information
Semi-classical Analysis of Schrödinger Equations
Semi-classical analysis is an asymptotic analysis where the Planck constant appearing in the Schrödinger equation is regarded as a small parameter. Under certain conditions, quantum mechanics is expected to approach classical mechanics in the semi-classical limit (Bohr’s correspondence principle). The asymptotic distribution of eigenvalues or resonances created by a bound or semi-bound state, respectively, is closely related to the existence and the geometry of “trapped” trajectories of the corresponding classical dynamics.
This problem is an extension of the famous question “Can one hear the shape of the drum?” (M. Kac), which examines the relationship between the geometry of a bounded domain and the asymptotic distribution of eigenvalues of its Dirichlet Laplacian. The useful WKB method consists of constructing an asymptotic power series solution globally with respect to the Planck constant. This power series diverges and the asymptotic form changes discontinuously when passing through turning points or caustics. This so-called Stokes phenomenon is a key to solve the above problem.