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

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

2003-03-13

arxiv.org/abs/math-ph/0303033v2

Rev. Math. Phys. 16 (2004), 559-582

Physics

Mathematical Physics

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

Scientific paper

10.1142/S0129055X04002084

We investigate a class of generalized Schr\"{o}dinger 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 Schr\"{o}dinger 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.

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