Schrödinger spectral kernels: High order asymptotic expansions

The large energy behavior of the spectral kernel for the N-body Schrödinger Hamiltonian is obtained. In a setting of a d-dimensional Euclidean space without boundaries, the Schrödinger Hamiltonian H is the sum of the negative Laplacian plus a real-valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N-body problem, since they do not necessarily decrease as |x|→∞. Let {e(x,y;λ):λεR} be the family of spectral kernels generated by H. In the λ→∞ limit, explicit higher order asymptotic expansions are obtained for e(x,y;λ) and its associated Riesz means. The asymptotic expansion is uniform in x and y and is accompanied by estimates of the error term. © 1983 American Institute of Physics.