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

Bound states for two dimensional Schrodinger equation with anisotropic interactions lambda r delta (rho - r) w (phi) localized on a circle of radius r are considered. lambda is a global parameter with energy as dimension. rho and phi are radial and angular coordinates. The Dirac distribution delta localizes the interaction on the circle. w (phi) measures the interaction at angle phi 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: w (phi) = cos (phi), lambda not equal 0. Second case: w (phi) = 1/(a + cos (phi)), a > 1, and lambda < 0. For the first case, the following results are obtained. Let the positive zeros j(nu,n) > 0 of Bessel function J nu (z) be numbered by integer n in increasing order, starting with n = 1 for the smallest zero. Define j(nu,0) = 0. Let j(1,l) and j(0,k) be the greatest values, which are smaller than vertical bar lambda vertical bar Mr(2)/(h) over bar (2), 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 -lambda Mr(2)/((h) over bar (2)root a(2) - 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. (C) 2015 AIP Publishing LLC.