Schrödinger operators with delta and delta'-potentials supported on hypersurfaces

Abstract

Self-adjoint Schrödinger operators with $\delta$ and ${\delta }^{\prime }$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman--Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten--von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with $\delta$ and ${\delta }^{\prime }$-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

to appear in Annales Henri Poincaré, 34 pages

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• One-dimensional Schroedinger operators with delta-prime-interactions on Cantor-type sets
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
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• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
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• Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction supported by a surface
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• Self-adjointness in Quantum Mechanics: a pedagogical path
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• A general review on the NLS equation with point-concentrated nonlinearity
• Limiting absorption principle and scattering matrix for Dirac operators with $\delta$-shell interactions
• Boundary triplets, tensor products and point contacts to reservoirs
• Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces
• Strong coupling asymptotics for $\delta$-interactions supported by curves with cusps
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• Scattering properties of two singularly interacting particles on the half-line
• Scattering of particles bounded to an infinite planar curve
• 2D Schrödinger operators with singular potentials concentrated near curves
• Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces
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• Trace Hardy inequality for the Euclidean space with a cut and its applications