Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces

Abstract

We show that a Schrödinger operator $A_{\delta ,\alpha }$ with a $\delta$-interaction of strength $\alpha$ supported on a bounded or unbounded $C^2$-hypersurface $\Sigma \subset {\mathbb{R}}^d$, $d\ge 2$, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator $A_{\delta ,\alpha }$ with a singular interaction is regarded as a self-adjoint realization of the formal differential expression $-\Delta -\alpha ⟨{\delta }_{\Sigma },\cdot ⟩{\delta }_{\Sigma }$, where $\alpha :\Sigma \to \mathbb{R}$ is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.

References in corpus (2)

• Bound states due to a strong $\delta$ interaction supported by a curved surface
• Schrödinger operators with $\delta$-interactions supported on conical surfaces

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