15-59 Jaroslav Dittrich, Pavel Exner. Christian K hn, Konstantin Pankrashkin

On eigenvalue asymptotics for strong $\delta$-interactions supported by surfaces with boundaries (262K, pdf) Jun 23, 15

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Abstract. Let $S\subset\mathbb{R}^3$ be a $C^4$-smooth relatively compact orientable surface with a sufficiently regular boundary. For $eta\in\RR_+$, let $E_j(eta)$ denote the $j$th negative eigenvalue of the operator associated with the quadratic form % -------------- % \[ H^1(\RR^3) i u\mapsto \iiint_{\mathbb{R}^3} | abla u|^2dx -eta \iint_S |u|^2d\sigma, \] % -------------- % where $\sigma$ is the two-dimensional Hausdorff measure on $S$. We show that for each fixed $j$ one has the asymptotic expansion % -------------- % \[ E_j(eta)=-\dfrac{eta^2}{4}+\mu^D_j+ o(1) \; ext{ as }\; eta o+\infty\,, \] % -------------- % where $\mu_j^D$ is the $j$th 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 $-\Delta_S$ 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 ${\mathcal O}(eta^{-1}\logeta)$.

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