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Approximation of Schrödinger operators with {\(\delta\)}-interactions supported on hypersurfaces. (English) Zbl 1376.35010

The paper under review considers Schrödinger operators with \(\delta\)-interactions supported on \(C^2\)-hypersurfaces in \(\mathbb{R}^d\) with \(d\geq 2\). It is shown that Schrödinger operators with suitable potentials supported on “thickened” layers of the hypersurface converge in norm resolvent sense to the Schrödinger operator with \(\delta\)-interaction on the hypersurface, where the norm of the difference of the resolvents is bounded by \(\varepsilon(1+|\ln \varepsilon|)\), where \(2\varepsilon\) is the thickness of the layer.
More precisely, let \(\Sigma\) be an orientable \(C^2\)-hypersurface and \(Q\in L_\infty(\mathbb{R}^d)\) be a background potential. Then there are two results:
(1) Let \(V\in L_\infty(\mathbb{R}^d)\) be supported on a thick layer around \(\Sigma\) and \(\alpha\) the average of \(V\) in transversal direction. Let \(V_\varepsilon\) be a suitably scaled version of \(V\) for the thick layer of thickness \(2\varepsilon\). Then \(-\Delta+Q-V_\varepsilon \to -\Delta+Q-\alpha\langle \delta_\Sigma,\cdot\rangle \delta_\Sigma\) in norm resolvent sense.
(2) Let \(\alpha\in L_\infty(\Sigma)\) and \(V\) a scaled constant extension of \(\alpha\) on a thick layer (scaling by thickness of the layer), and \(V_\varepsilon\) as above. Then \(-\Delta+Q-V_\varepsilon \to -\Delta+Q-\alpha\langle \delta_\Sigma,\cdot\rangle \delta_\Sigma\) in norm resolvent sense.
Both convergences are proved with the above rate.
Main tools for proving the results are fine integral estimates for Green functions and a perturbation argument to reduce the problem to the case \(Q=0\).

MSC:

35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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