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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].

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
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