Point Interactions with an Internal Structure as Limits of Nonlocal Separable Potentials

Abstract

A one-parameter family of self-adjoint extensions of the symmetric operator h₀=−Δ in L₂(R³) acting on the space of smooth functions which vanish in the vicinity of the origin serves as a rigorous definition for one-particle point interaction Hamiltonian [1]. The resolvents R^{( α )} of this family h^{( α )} can be given explicitly and in the p-representation they have the form

$${{R}^{(\alpha )}}(z)={{R}_{0}}(z)-t(z)K(z),\operatorname{Im}z\ne 0,$$

(1)

where R₀(z)=(p²−z)⁻¹ is the resolvent of the Laplace operator, K(z) is given by the integral kernel

$$K(p,k,z)={{({{p}^{2}}-z)}^{-1}}{{({{k}^{2}}-z)}^{-1}},$$

and $t(z)={{(2{{\pi }^{2}})}^{-1}}{{(-\sqrt{-z}+\alpha )}^{-1}},\operatorname{Re}(\sqrt{-z})\ge 0$ for z<0, plays the role of the t-matrix if α∈R; on the other hand, α=∞ corresponds to the free operator −Δ.