Spectral asymptotics induced by approaching and diverging planar circles

Abstract

We consider two dimensional system governed by the Hamiltonian with delta interaction supported by two concentric circles separated by distance d. We analyze the asymptotics of the discrete eigenvalues for d → 0 as well as for d → ∞.

Rank one perturbations supported by hybrid geometries and their deformations

“…If the curve is planar, similarly if the surface is embedded in R 3 (with some regularity conditions on the curves/surfaces), one way is to interpret these interactions expressed by the following quadratic forms ∫ R 2 |∇ψ| 2 d 2 x − λ ∫ Γ |ψ| 2 ds and ∫ R 3 |∇ψ| 2 d 3 x − λ ∫ Σ |ψ| 2 dA, respectively, and then prove that there exist associated self-adjoint Hamiltonians. [10][11][12][13][14] Other ways are to impose proper boundary conditions (continuity and jump discontinuity conditions at Γ; see Remark 4.1 in Refs. 10 and 15) or employ scaled potentials 16 or direct construction of the resolvent 10,17,18 (see also Ref.…”