Strong-coupling asymptotic expansion for Schrödinger operators with a singular interaction supported by a curve in ${\mathbb{R}}^3$

Abstract

We investigate a class of generalized Schrödinger operators in $L^2({\mathbb{R}}^3)$ with a singular interaction supported by a smooth curve $\Gamma$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when $\Gamma$ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of $\Gamma$. In the same way we obtain an asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if $\Gamma$ is not a straight line and the singular interaction is strong enough.

LaTeX 2e, 30 pages; minor improvements, to appear in Rev. Math. Phys

References in corpus (4)

• Geometrically induced spectrum in curved leaky wires
• Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop
• Curvature-induced bound states for a $\delta$ interaction supported by a curve in ${\mathbb{R}}^3$
• Band gap of the Schroedinger operator with a strong delta-interaction on a periodic curve

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• Singular interactions supported by embedded curves
• A Klein Gordon Particle Captured by Embedded Curves
• A straight waveguide with a wire inducing resonances
• Infinitely many singular interactions on noncompact manifolds