BEHAVIOR OF SPECTRA OF SCHRODINGER-OPERAT ORS AT LOW-TEMPERATURES

On a finite graph M, consider the generator L(beta), at the temperatur e beta(-1) greater than or equal to 0, of the simulated annealing algo rithm associated to a potential U. Let 0 = lambda(0)(beta) > lambda(1) (beta) greater than or equal to ... greater than or equal to lambda(N- 1)(beta) be its eigenvalues. We extend a result of HOLLEY and STROOCK on the asymptotics of lambda(1)(beta), by showing that for 1 less than or equal to i less than or equal to N - 1, there exists constants b(i ) greater than or equal to a(i) > 0 (depending only on M) and c(i) (M, U) greater than or equal to 0, such that for all beta greater than or equal to 0, a(i) exp (-c(i) (M, U)beta) less than or equal to - lambd a(i) (beta) less than or equal to b(i) exp (-c(i) (M, U)beta), and we give a geometrical interpretation of c(i) (M, U). Furthermore, let (L) over tilde(beta) = g(beta)(-1) circle L(beta) circle g(beta), where g (beta) = exp (beta/2 U), be the Schrodinger operator associated to L(b eta), we shall prove the convergence of the sum of the eigenprojection s (of (L) over tilde(beta)) corresponding to the eigenvalues lambda(i) (beta) for which limp(beta-->+infinity) beta(-1) ln(lambda(i)(beta)) = -k (for any k is an element of {c(1) (M, U), ..., c(N-1)(M, U)} fixed ) and identify the limit. Then we will generalize this kind of results to the continuous case where L(beta). = Delta .- -beta(del U, del.) o n a compact and connected Riemannian manifold M, by using the asymptot ics of lambda(1)(beta) given by HOLLEY, KUSUOKA and STROOCK.