Bouas, J. D.; Fulling, S. A.; Mera, F. D.; Thapa, K.; Trendafilova, C. S.; Wagner, J. Investigating the spectral geometry of a soft wall. (English) Zbl 1317.81095 Barnett, Alex H. (ed.) et al., Spectral geometry. Based on the international conference, Dartmouth, NH, USA, July 19–23, 2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5319-1/hbk). Proceedings of Symposia in Pure Mathematics 84, 139-154 (2012). Summary: 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, \(z^\alpha\). We report investigations of this model with considerable student involvement. An exact analytical solution with some numerics for \(\alpha=1\) and an asymptotic (semiclassical) analysis of a related problem for \(\alpha=2\) are presented.For the entire collection see [Zbl 1253.58001]. Cited in 3 Documents MSC: 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81V99 Applications of quantum theory to specific physical systems 35P05 General topics in linear spectral theory for PDEs 81T99 Quantum field theory; related classical field theories PDFBibTeX XMLCite \textit{J. D. Bouas} et al., Proc. Symp. Pure Math. 84, 139--154 (2012; Zbl 1317.81095) Full Text: DOI arXiv