Integral Equations and Operator Theory

, Volume 75, Issue 3, pp 341–362

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

Article

DOI: 10.1007/s00020-012-2027-z

Cite this article as:
Golovaty, Y. Integr. Equ. Oper. Theory (2013) 75: 341. doi:10.1007/s00020-012-2027-z

Abstract

For real \({L_\infty(\mathbb{R})}\)-functions \({\Phi}\) and \({\Psi}\) of compact support, we prove the norm resolvent convergence, as \({\varepsilon}\) and \({\nu}\) tend to 0, of a family \({S_{\varepsilon \nu}}\) of one-dimensional Schrödinger operators on the line of the form
$$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$
provided the ratio \({\nu/\varepsilon}\) has a finite or infinite limit. The limit operator S0 depends on the shape of \({\Phi}\) and \({\Psi}\) as well as on the limit of ratio \({\nu/\varepsilon}\). If the potential \({\alpha\Phi}\) possesses a zero-energy resonance, then S0 describes a non trivial point interaction at the origin. Otherwise S0 is the direct sum of the Dirichlet half-line Schrödinger operators.

Mathematics Subject Classification (2010)

34L40 34B09 81Q10 

Keywords

1D Schrödinger operator resonance short range interaction point interaction δ-potential δ′-potential distributional potential solvable model norm resolvent convergence 

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Differential EquationsIvan Franko National University of LvivLvivUkraine

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