Schrödinger semigroups for vector fields

Abstract

Suppose H is the Hamiltonian that generates time evolution in an N‐body, spin‐dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r×r matrix operator given by the sum of the negative Laplacian −Δx on the (d=3N)‐dimensional Euclidean space Rd plus a Hermitian local matrix potential W(x). Uniform higher‐order asymptotic expansions are derived for the time‐evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix‐valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex‐valued matrix measures. This class is suitable for the N‐body problem since interactions of this type do not necessarily decrease as ‖x‖→∞. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup {e−zH‖Re z>0}. The asymptotic expansions obtained are valid for all finite coordinate space dimensions d and all finite vector space dimensions r, and are uniform in Rd×Rd. The order of expansion is solely a function of the smoothness properties of the local potential W(x).