TITLE:  Upper Bounds for the Eigenvalue Counting Function of the Laplacian and Schrodinger Operators

 

ABSTRACT:  TheWeyl asymptotics provide information about the asymptotic distribution of the eigenvalues of the Laplacian in a bounded domain. Polya conjectured that the Weyl asymptotics actually provide an upper bound for the eigenvalue counting function. In the late 70's, Cwickel, Rosenbljum and Lieb independently (and with dierent techniques) proved an upper bound on the number of negative eigenvalues of a Schrodinger operator on Rd. I will give a motivated presentation of Lieb's technique, which uses path integrals and the Feynman- Kac formula. Lieb also used this bound to get a bound proportional to Polya's conjecture.