Rank One Perturbations Supported by Hybrid Geometries and Their Deformations

Abstract

We study the hybrid type of rank one perturbations in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$, where the perturbation supported by a circle/sphere is considered together with the delta potential supported by a point outside of the circle/sphere. The construction of the self-adjoint Hamiltonian operator associated with the formal expressions for the rank one perturbation supported by a circle and by a point is explicitly given. The bound state energies and scattering properties for each problem are also studied. Finally, we consider the rank one perturbation supported by a deformed circle/sphere and show that the first order change in the bound state energies under small deformations of the circle/sphere has a simple geometric interpretation. Finally, we consider the delta potentials supported by deformed circle/sphere and show that the first order change in the bound state energies under small deformations of circle/sphere has a simple geometric interpretation.

arXiv comment31 pages, 14 figures, typos corrected, new references added, title changed due to the suggestion of the referee, published version

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• Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal ${\delta }^{\prime }$ interactions