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Schrödinger operators with \((\alpha \delta'+\beta \delta)\)-like potentials: norm resolvent convergence and solvable models. (English) Zbl 1265.34320

Let \(\Phi,\Psi\) be integrable compactly supported real functions on \(\mathbb R\). The author proves the norm resolvent convergence, as \(\varepsilon \to 0\), of a family \(S_\varepsilon\) of one-dimensional Schrödinger operators on \(\mathbb R\), of the form \[ S_\varepsilon =-\frac{d^2}{dx^2}+\alpha \varepsilon^{-2}\Phi (\varepsilon^{-1}x)+\beta \varepsilon^{-1}\Psi (\varepsilon^{-1}x). \] If the equation \(-u''+\alpha \Phi (x)u=0\) possesses a bounded solution on \(\mathbb R\), then the limit of \(S_\varepsilon\) can be interpreted as a realization of the formal Hamiltonian \[ -\frac{d^2}{dx^2}+\alpha \delta' (x)+\beta \delta (x). \] Otherwise \(S_\varepsilon\) tends to the direct sum \(S_-\oplus S_+\) of the Dirichlet half-line Schrödinger operators \(S_\pm\).

MSC:

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