Bound states due to a strong $\delta$ interaction supported by a curved surface

Abstract

We study the Schrödinger operator $-\Delta -\alpha \delta (x-\Gamma )$ in $L^2({\mathbb{R}}^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a uniformly elliptic metric. We show that if $\Gamma$ is asymptotically planar in a suitable sense and $\alpha >0$ is sufficiently large this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a ``two-dimensional'' comparison operator determined by the geometry of the surface $\Gamma$. [A revised version, to appear in J. Phys. A]

LaTeX 2e, 21 pages

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Cited by in corpus (19)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces
• Generalized interactions supported on hypersurfaces
• Spectral estimates for resolvent differences of self-adjoint elliptic operators
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• Eigenvalue asymptotics for the Schrödinger operator with a $\delta$-interaction on a punctured surface
• Soft quantum waveguides in three dimensions
• Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction supported by a surface
• Scattering from local deformations of a semitransparent plane
• Compact submanifolds supporting singular interactions
• Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space
• Lower bound on the spectrum of the Schrödinger operator in the plane with delta-potential supported by a curve
• Random Schr"odinger operators on manifolds
• Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions
• Rank One Perturbations Supported by Hybrid Geometries and Their Deformations
• Infinitely many singular interactions on noncompact manifolds