Michael Taylor
University of North Carolina, Chapel Hill
If V is a non-negative, locally integrable function on Euclidean space R^n, then L = - Laplacian + V defines a positive, self-adjoint operator on L^2(R^n). It is classical that this operator has purely discrete spectrum provided V(x) tends to + infinity as |x| goes to infinity. A. Molchanov produced a necessary and sufficient condition that L have purely discrete spectrum, in terms of integrals of V over various cubes, with certain "negligible" sets omitted. This result was improved in recent work of V. Maz'ya and M. Shubin. We will discuss an alternative formulation of a necessary and sufficient condition for discreteness, in terms of "scattering length." This quantity, introduced by M. Kac, is associated to a positive integrable function. It extends in a natural way the concept of capacity, which is defined on sets. There is a formula for scattering length in terms of Wiener measure, which invites one to estimate eigenvalues by considering how Brownian paths interact with positive potentials.