Strong coupling asymptotics for δ-interactions supported by curves with cusps

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:491

Article

Flamencourt, Brice¹  Pankrashkin, Konstantin² 

[1] Univ Paris Saclay, Lab Math Orsay, CNRS, F-91405 Orsay, France

[2] Carl von Ossietzky Univ Oldenburg, Inst Math, D-26111 Oldenburg, Germany

关键词: Schrodinger operator;    delta-Potential;    Boundary with a cusp;    Eigenvalue asymptotics;    Dimension reduction;    Effective operator;   

DOI  :  10.1016/j.jmaa.2020.124287

来源: Elsevier

【 摘 要 】

Let Gamma subset of R-2 be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve vertical bar x(2)vertical bar = x(1)(p) for some p > 1. We study the eigenvalues of the Schrodinger operator H-alpha with the attractive delta-potential of strength alpha > 0 supported by Gamma, which is defined by its quadratic form H-1(R-2) (sic) u bar right arrow integral integral(R2) vertical bar del u vertical bar(2) dx -alpha integral(Gamma) u(2) ds, where ds stands for the one-dimensional Hausdorff measure on Gamma. It is shown that if n is an element of N is fixed and a is large, then the well-defined nth eigenvalue E-n (H-alpha) of H-alpha behaves as E-n(H-alpha) = -alpha(2) + 2(2/p+2)epsilon(n) alpha(6/p+2) +O(alpha(6/p+2-eta)), where the constants epsilon(n) > 0 are the eigenvalues of an explicitly given one-dimensional Schrodinger operator determined by the cusp, and eta > 0. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when Gamma is smooth or piecewise smooth with non-zero angles. (C) 2020 Elsevier Inc. All rights reserved.

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