1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials. (English) Zbl 1270.34198

For real \({L_\infty(\mathbb{R})}\)-functions \({\Phi}\) and \({\Psi}\) of compact support, the author 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 \(S _{0}\) depends on the shape of \({\Phi}\) and \({\Psi}\) as well as on the limit of the ratio \({\nu/\varepsilon}\). If the potential \({\alpha\Phi}\) possesses a zero-energy resonance, then \(S _{0}\) describes a nontrivial point interaction at the origin. Otherwise \(S _{0}\) is the direct sum of Dirichlet half-line Schrödinger operators.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B09 Boundary eigenvalue problems for ordinary differential equations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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