1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials

Details 1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials 1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials

2012-02-21

arxiv.org/abs/1202.4711v2

Mathematics

Spectral Theory

17 pages

Scientific paper

For real bounded functions \Phi and \Psi of compact support, we prove the norm resolvent convergence, as \epsilon and \nu tend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{\epsilon, \nu}= -D^2+\alpha\epsilon^{-2}\Phi(\epsilon^{-1}x)+\beta\nu^{-1}\Psi(\nu^{-1}x), provided the ratio \nu/\epsilon has a finite or infinity limit. The limit operator S_0 depends on the shape of \Phi and \Psi as well as on the limit of ratio \nu/\epsilon. If the potential \alpha\Phi possesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Otherwise S_0 is the direct sum of the Dirichlet half-line Schroedinger operators.

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