Spectral asymptotics for ${\delta }^{\prime }$ interaction supported by a infinite curve

Abstract

We consider a generalized Schrödinger operator in $L^2({\mathbb{R}}^2)$ describing an attractive ${\delta }^{\prime }$ interaction in a strong coupling limit. ${\delta }^{\prime }$ interaction is characterized by a coupling parameter $\beta$ and it is supported by a $C^4$-smooth infinite asymptotically straight curve $\Gamma$ without self-intersections. It is shown that in the strong coupling limit, $\beta \to 0_{+}$, the eigenvalues for a non-straight curve behave as $-\frac{4}{{\beta }^2}+{\mu }_j+\mathcal{O}(\beta |\ln \beta |)$, where ${\mu }_j$ is the $j$-th eigenvalue of the Schrödinger operator on $L^2(\mathbb{R})$ with the potential $-\frac{1}{4}{\gamma }^2$ where $\gamma$ is the signed curvature of $\Gamma$.

References in corpus (2)

• Leaky Quantum Graphs: A Review
• Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction supported by a surface

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