h(cross)-smooth quantum solutions: a semiclassical method

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Abstract
The h(cross) to 0 semiclassical expansion of Wigner (1932) and Kirkwood (1933) is obtained for quantum systems of finitely many particles. In a d-dimensional Euclidean space Rd without boundaries quantum systems defined by a Hamiltonian, H, given as the sum of the negative Laplacian perturbed by a potential nu (x) are considered. The semiclassical behaviour of the kernels of the semigroup family of operators (e-zH:Rez>or=0) is determined in terms of an asymptotic expansion in the variable h(cross). The order of the expansion is proportional to the number of bounded derivatives nu (x) supports. The expansion is uniform in Rd*Rd and accompanied by explicit bounds for the error term. The results are obtained for potentials nu (x) that are Fourier images of complex bounded measures.

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