A TRACE FORMULA FOR MULTIDIMENSIONAL SCHRODINGER-OPERATORS

We prove multidimensional analogs of the trace formula obtained previo usly for one-dimensional Schrodinger operators. For example, let V be a continuous function on [0, 1](v) subset of R(v). For A subset of {1, ..., v}, let -Delta(A), be the Laplace operator on [0, 1](v) with mixe d Dirichlet-Neumann boundary conditions phi(x) = 0, x(j) = 0 or x(j) = 1 for j is an element of A, partial derivative phi/partial derivative (j) = 0, x(j) = 0 or x(j) = 1 for j is not an element of A. Let \A\ = number of points in A. Then we'll prove that Tr((A subset of{1,...,v}) Sigma (-1)(\A\)e(-t(-Delta A + V)) = 1 - t [V] + o(t) as t down arrow 0 with [V] the average of V at the 2(v) corners of [0, 1](v). (C) 1996 Academic Press, Inc.