A Laplace operator with boundary conditions singular at one point
Artikel i vetenskaplig tidskrift, 2009
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.