Bound states for two dimensional Schrödinger equation with anisotropic interactions localized on a circle

Abstract

Bound states for two dimensional Schrödinger equation with anisotropic interactions λrδρ−rwφ localized on a circle of radius r are considered. λ is a global parameter with energy as dimension. ρ and φ are radial and angular coordinates. The Dirac distribution δ localizes the interaction on the circle. wφ measures the interaction at angle φ on the circle. A general method for determination of energies, mean values of different operators, normalized wave functions both in configuration space and momentum space is given. This method is applied to two cases. First case: wφ=cosφ, λ ≠ 0. Second case: wφ=1/a+cosφ, a > 1, and λ < 0. For the first case, the following results are obtained. Let the positive zeros jν,n > 0 of Bessel function Jνz be numbered by integer n in increasing order, starting with n = 1 for the smallest zero. Define jν,0 = 0. Let j1,ℓ and j0,k be the greatest values, which are smaller than λMr2/ħ2, with M the mass. Then, the dimension of the vector space generated by even bound states is ℓ + 1, and the one generated by odd bound states is k. For the second case, let k be the greatest positive or zero integer, which is smaller than −λMr2/ħ2a2−1. Then, the dimension of the vector space generated by even bound states is k + 1, and the one generated by odd bound states is k.