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

Golovaty, Yuriy

Methods Funct. Anal. Topol. 18, No. 3, 243-255 (2012).

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 \(S_\pm\).

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

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

Keywords:

one-dimensional Schrödinger operators; point interaction; \(\delta\)-potential; \(\delta'\)-potential