Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3

Details Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3 Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3

2007-12-03

arxiv.org/abs/0712.0313v1

Physics

Mathematical Physics

LaTeX, 29 pages

Scientific paper

10.1063/1.2845419

We consider Schr\"odinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a finite curve $\Gamma$. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if $\Gamma$ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length $2\epsilon$. We derive an asymptotic expansion with the leading term which a multiple of $\epsilon \ln\epsilon$.

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