## 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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