Asymptotics of determinants of discrete Schrodinger operators

We consider the asymptotics of the determinants of large discrete Schrodinger operators, i.e. "discrete Laplacian + diagonal": T-n(f) = -[delta(j,j+1) + delta(j+1,j)] + diag(f(1/n), f(2/n), . . . , f(n/n)) We extend a result of M. Kac [3] who found a formula for lim(n ->infinity)det(T-n(f))/G(f)(n) in terms of the values of f, where G(f) is a constant. We extend this result in two ways: First, we consider shifting the index: Let T-n(f, epsilon) = -[delta(j,j+1), + delta(j+1,j)] + diag(f(epsilon/n), f(1 + epsilon/n), . . . , f(n - 1 + epsilon/n)). We calculate lim det T-n(f ; epsilon)/G(f)(n) and show that this limit can be adjusted to any positive number by shifting epsilon, even though the asymptotic eigenvalue distribution of T-n(f ; epsilon) does not depend on epsilon. Secondly, we derive a formula for the asymptotics of det T-n(f)/G(f)(n) when f has jump discontinuities. In this case the asymptotics depend on the fractional part of cn, where c is a point of discontinuity.