On eigenvalue asymptotics for strong delta-interactions supported by surfaces with boundaries

0458924 - ÚJF 2017 RIV NL eng J - Článek v odborném periodiku
Dittrich, Jaroslav - Exner, Pavel - Kuhn, C. - Pankrashkin, K.
On eigenvalue asymptotics for strong delta-interactions supported by surfaces with boundaries.
Asymptotic Analysis. Roč. 97, 1-2 (2016), s. 1-25. ISSN 0921-7134. E-ISSN 1875-8576
Grant CEP: GA ČR(CZ) GA14-06818S
Institucionální podpora: RVO:61389005
Klíčová slova: singular Schrodinger operator * delta-interaction * strong coupling * eigenvalue
Kód oboru RIV: BE - Teoretická fyzika
Impakt faktor: 0.933, rok: 2016

Let S subset of R-3 be a C-4-smooth relatively compact orientable surface with a sufficiently regular boundary. For beta is an element of R+, let E-j(beta) denote the jth negative eigenvalue of the operator associated with the quadratic form
H-1(R-3) (sic) u (sic) integral integral integral(R3) vertical bar del u vertical bar(2) dx - beta integral integral(s) vertical bar u vertical bar(2) d sigma where sigma is the two-dimensional Hausdorff measure on S. We show that for each fixed j one has the asymptotic expansion E-j(beta) = -beta(2)/4 + mu(D)(j) + o(1) as beta -> +infinity where mu(D)(j) is the jth eigenvalue of the operator -Delta s +K - M-2 on L-2 (S), in which K and M are the Gauss and mean curvatures, respectively, and As is the Laplace Beltrami operator with the Dirichlet condition at the boundary of S. If, in addition, the boundary of S is C-2-smooth, then the remainder estimate can be improved to O(beta(-1) log beta).
Trvalý link: http://hdl.handle.net/11104/0259134