Affiliations: [a] Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, Řež near Prague, Czechia | [b] Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Prague, Czechia. E-mails: dittrich@ujf.cas.cz, exner@ujf.cas.cz | [c] Institut für Numerische Mathematik, Technische Universität Graz, Graz, Austria. E-mail: kuehn@tugraz.at | [d] Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Orsay, France. E-mail: konstantin.pankrashkin@math.u-psud.fr
Correspondence:
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Corresponding author: Konstantin Pankrashkin, Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France. E-mail: konstantin.pankrashkin@math.u-psud.fr.
Abstract: Let S⊂R3 be a C4-smooth relatively compact orientable surface with a sufficiently regular boundary. For β∈R+, let Ej(β) denote the jth negative eigenvalue of the operator associated with the quadratic form
H1(R3)∋u↦∭R3|∇u|2dx−β∬S|u|2dσ,
where σ is the two-dimensional Hausdorff measure on S. We show that for each fixed j one has the asymptotic expansion
Ej(β)=−β24+μjD+o(1)as β→+∞,
where μjD is the jth eigenvalue of the operator −ΔS+K−M2 on L2(S), in which K and M are the Gauss and mean curvatures, respectively, and −ΔS is the Laplace–Beltrami operator with the Dirichlet condition at the boundary of S. If, in addition, the boundary of S is C2-smooth, then the remainder estimate can be improved to O(β−1logβ).
