### Abstract

Bound states for two dimensional Schrödinger equation with anisotropic interactions
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.
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:
, λ ≠ 0. Second case:
, a > 1, and λ < 0. For the first case, the following results are obtained. Let the positive zeros j
ν,n
> 0 of Bessel function
be numbered by integer n in increasing order, starting with n = 1 for the smallest zero. Define j
ν,0 = 0. Let j
1,ℓ and j
0,k
be the greatest values, which are smaller than
, 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
. 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.

Received 10 June 2014
Accepted 22 January 2015
Published online 10 February 2015

Article outline:

I. INTRODUCTION
II. GENERAL THEORY
A. The Hamiltonian *H*
_{
λ
}
B. The resolvent *G*
_{
λ
}(*z*)
C. The bound states
D. General properties of the *C*(*w*) matrices
III. APPLICATIONS OF THE GENERAL THEORY TO TWO PARTICULAR CASES
A. A non invertible continuous interaction
B. An invertible continuous interaction
IV. CONCLUDING REMARKS

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