Domain perturbations, shift of eigenvalues and capacity

The notion of capacity of a subspace which was introduced in [16] is used t o prove new estimates on the shift of the eigenvalues which arises if the f orm domain of a self-adjoint and semibounded operator is restricted to a sm aller subspace. The upper bound on the shift of the spectral bound given in [16] is improved and another lower bound is proved which leads to a genera lization of Thirring's inequality if the underlying Hilbert space is an L-2 -space. Moreover we prove a similar capacitary upper bound for the second e igenvalue. The results are applied to elliptic constant coefficient differe ntial operators of arbitrary order. Finally it is given a capacitary charac terization for the shift of the spectral bound being positive which works f or operators with spectral bound of arbitrary type. (C) 2000 Academic Press .