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Home  >  Volume 35

On the Dimensions of the Vector Space Generated by the Bound States for a Two Dimensional SchrödingerEquation with Anisotropic Interactions Confined in a Riemann Surface E2 by E.O. Ifidonand L. A. Umwerhiaye (pages 5-10)
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Abstract

In this paper, we consider the dimensions of the vector space generated by the bound states for a 2-dimensional Schrödinger equation with an anisotropic potential∑_(j=1)^N▒〖v_j (r)ω(ϕ)δ(r-a_j)〗 bounded by two-dimensional Riemann surfaces of radius aj, where δ is the usual Kronecker delta function. The variables r and ϕ are radial and angular coordinates respectively, ω(ϕ) measures the interaction at angle ϕ on the Riemann surface. The Hamiltonian is the sum of the kinetic and the interaction part. The interactions are separable and are centered at arbitrary points on the surface. The dimension of the vector space generated by bound states is obtained using a general method suitable for the determination of energies and mean values of different operators corresponding to the normalized wave function. Conditions for the existence and for the number of bound states in finite linear chains are formulated in terms of the parameters of the interactions and intercentre distances. It is shown that for the model considered in this paper, the dimension of the vector space generated by the bound states reduces as the number of inter-particle interactions N in the surface increases.

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