Rank One Perturbations Supported by Hybrid Geometries and Their Deformations

TL;DR

This paper investigates rank one perturbations supported by geometric shapes like circles and spheres, analyzing their self-adjoint Hamiltonians, bound states, and scattering, with a focus on how small deformations affect bound state energies.

Contribution

It explicitly constructs the Hamiltonians for rank one perturbations supported by circles and spheres, including deformations, and interprets the first-order energy changes geometrically.

Findings

Explicit Hamiltonian construction for supported perturbations.

Analysis of bound state energies and scattering properties.

First-order energy change under deformation has a simple geometric interpretation.

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.