A Laplace operator with boundary conditions singular at one point

Marletta, Marco ORCID: https://orcid.org/0000-0003-1546-4046 and Rozenblum, Grigori 2009. A Laplace operator with boundary conditions singular at one point. Journal of Physics A: Mathematical and Theoretical 42 (12) , 125204. 10.1088/1751-8113/42/12/125204

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Abstract

We discuss a recent paper of Berry and Dennis (J. Phys. A: Math. Theor. 2008 41 135203) concerning a Laplace operator on a smooth domain with singular boundary condition. We explain a paradox in the article (J. Phys. A: Math. Theor. 2008 41 135203) and show that if a certain additional condition is imposed, the result is a spectral problem for a self-adjoint operator having only eigenvalues and no continuous spectrum. The eigenvalues accumulate at ±∞ only, and we obtain the asymptotic behaviours of the counting functions n+(λ) and n−(λ) for positive and negative eigenvalues. The physical meaning of the additional boundary condition is not yet clear.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Publisher:	IOP
ISSN:	1751-8113
Last Modified:	18 Oct 2022 13:27
URI:	https://orca.cardiff.ac.uk/id/eprint/13969

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