2003/01/15 by Pavel Exner, Exner, Pavel
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #cond-mat #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.math-ph/0301021
AMSTeX, 12 pages; to appear in Proceedings of the NSF Summer Research Conference (Mt. Holyoke 2002); AMS "Contemporary Mathematics" Series, Providence, R.I., 2003
arxiv created 2003/01/15 · arxiv updated 2009/11/30
We investigate the operator -Δ-αδ(x-Γ) in L2(ℝ3), where Γ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as α→∞ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface Γ. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic Γ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.