Time decay and spectral kernel asymptotics

For quantum systems in ℝ³ defined by a Hamiltonian H given as the sum of the negative Laplacian perturbed by a real-valued potential v(x), the large time behavior of the fundamental solution of the time-dependent Schrödinger equation is investigated. For a suitably restricted class of potentials that have algebraic decay as |x|→∞, the continuous spectrum portion of the fundamental solution is characterized by an asymptotic expansion as t→ ± ∞, which is uniform in compact subsets of ℝ³ X ℝ³. These results are then applied to derive the large energy asymptotic expansions of the spectral kernel associated with H. © 1985 American Institute of Physics.