Investigating the Spectral Geometry of a Soft Wall

Abstract

The idealized theory of quantum vacuum energy density is a beautiful application of the spectral theory of differential operators with boundary conditions, but its conclusions are physically unacceptable. A more plausible model of a reflecting boundary that stays within linear spectral theory confines the waves by a steeply rising potential function, which can be taken as a power of one coordinate, z^α. We report investigations of this model with considerable student involvement. An exact analytical solution with some numerics for α=1 and an asymptotic (semiclassical) analysis of a related problem for α=2 are presented.

16 pages,4 figures; International Conference on Spectral Geometry (Dartmouth, 2010). Revision (final submission to Proc. Symp. Pure Math.) has minor updates and corrections