We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theory on the trace ideals, c#~(p < oo). 1.

Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random

We investigate the two-dimensional scattering of a point-particle from n non-overlapping fixed disks and study -- using these systems -- the connection between the spectral properties of the quantum mechanical scattering matrix and its semi-classical equivalent based on the Gutzwiller-Voros zetafunction. We rewrite the determinant of the scattering matrix in such a way that it separates into the product over n determinants of 1-disk scattering matrices -- representing the incoherent part of the scattering from the n disk system -- and the ratio of two mutually complex conjugate determinants of the genuinely multi-scattering kernel, M, of KKR-type which represents the coherent part of the n-disk scattering. Our result is well-defined at every step of the calculation, as the on-shell T--matrix and the kernel A = M\Gamma1 are shown to be trace-class. We stress that the cumulant expansion (which defines the determinant over an infinite, but trace class matrix) imposes the curvature regula...

d either by its series expansion, or as a limit of an innite product: e A = 1 X k=0 1 k! A k ; A 0 = 1 (F.5) = lim N!1 1 + 1 N A N (F.6) The rst equation follows from the second one by the binomial theorem, so these indeed are equivalent denitions. That the terms of order O(N 2 ) or smaller do not matter follows from the bound 1 + x N N < 1 + x + xN N N < 1 + x + N N ; where jx N j < . If lim<F8.928

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Abstract. The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is self– adjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the operator. First we estimate upper and lower bounds for the ε–capacity (and then for the metric information), and explicit computations in some specific cases are given; then the problem is reformulated from a probabilistic viewpoint and use is made of the probabilistic information theory. The results obtained by these two approaches are then compared. 1.