We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a selfaveraging integrated density of states and a Subin type trace formula.

Abstract. In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L 2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the ”principle of not feeling the boundary ” (due to M. Kac), for heat kernels associated to boundary value problems.

We propose a new method to analyze and represent data recorded on a domain of gen-eral shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonal-ize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neu-mann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the compu-tation. We also show that our method is better suited for small sample data than the Karhunen-Loève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further ap-plication, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain. Key words: Laplacian eigenfunctions, boundary conditions, Green’s function, spectral decomposition, Karhunen-Loève transform, principal component analysis, heat equation