  In this paper we extend classical criteria for determining lower bounds for
the least point of the essential spectrum of second-order elliptic differential
operators on domains $\Omega\subset\R^n$ allowing for degeneracy of the
coefficients on the boundary. We assume that we are given a sesquilinear form
and investigate the degree of degeneracy of the coefficients near
$\partial\Omega$ that can be tolerated and still maintain a closable
sesquilinear form to which the First Representation Theorem can be applied.
Then, we establish criteria characterizing the least point of the essential
spectrum of the associated differential operator in these degenerate cases.
Applications are given for convex and non-convex $\Omega$ using Hardy
inequalities, which recently have been proven in terms of the distance to the
boundary, showing the spectra to be purely discrete.

Spectral Theory