Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction on a planar loop

Abstract

We consider a generalized Schrödinger operator in $L^2({\mathbb{R}}^2)$ with an attractive strongly singular interaction of ${\delta }^{\prime }$ type characterized by the coupling parameter $\beta >0$ and supported by a $C^4$-smooth closed curve $\Gamma$ of length $L$ without self-intersections. It is shown that in the strong coupling limit, $\beta \to 0_{+}$, the number of eigenvalues behaves as $\frac{2L}{πβ} + \OO(|\lnβ|)$, and furthermore, that the asymptotic behaviour of the $j$-th eigenvalue in the same limit is $-\frac{4}{β^2} +μ_j+\OO(β|\lnβ|)$, where ${\mu }_j$ is the $j$-th eigenvalue of the Schrödinger operator on $L^2(0,L)$ with periodic boundary conditions and the potential $-\frac{1}{4}{\gamma }^2$ where $\gamma$ is the signed curvature of $\Gamma$.

References in corpus (1)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces

Cited by in corpus (4)

• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• On absence of bound states for weakly attractive ${\delta }^{\prime }$-interactions supported on non-closed curves in ${\mathbb{R}}^2$
• Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction supported by a surface
• Trace Hardy inequality for the Euclidean space with a cut and its applications