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arXiv · 1207.2271

Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc

Pavel Exner·Konstantin Pankrashkin
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Abstract

We consider a singular Schrödinger operator in $L^2(\mathbb{R}^2)$ written formally as $-Δ- βδ(x-γ)$ where $γ$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the $j$th negative eigenvalue of this operator behaves in the strong-coupling limit, $β\to +\infty$, asymptotically as \[ E_j(β)=-\frac{β^2}{4} +μ_j +\mathcal{O}\Big(\dfrac{\logβ}β\Big), \] where $μ_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{κ(s)^2}{4}\, \] on $L^2(0,L)$ with $κ(s)$ being the signed curvature of $γ$ at the point $s\in(0,L)$.

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Pavel Exner, Konstantin Pankrashkin. 2014-10-31. Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc. https://doi.org/10.1080/03605302.2013.851213

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