Abstract. We have studied bound states of the Schrödinger equation for an attractive potential with any finite number (P) of Dirac delta-functions in Rn where n = 1, 2, 3, .... The potential is radially symmetric for n ≥ 2 and is given as V(r) = −2/2m ∑Pi = 1 σiδ(r − ri) where σi > 0, r1 < r2 < &cdots; < rP, and ri (0, +∞) for n ≥ 2, ri (−∞, +∞) for n = 1. By separating angular degrees of freedom, the radial equation is obtained for n ≥ 2 and applications of the boundary conditions lead to P transfer matrices which are used to form an equation for the eigenvalues. We have proven that, for given n and l, the bound state solutions of the radial equation are non-degenerate and there are at most P bound state solutions of the radial equation and hence P bound state energy levels for a potential with P attractive Dirac delta-functions. Given l and n ≥ 2, for P = 1, we have shown that there exists one and only one solution of the radial equation if σ1 r1 > 2l + n − 2 and none otherwise. We have also proven that there are at most P positive roots for the equation X22(k) = 0 where X = (X11X21X12X22) = MPMP−1 ... M1 and Mi SL(2, R) are the particular transfer matrices mentioned above.

PACS number: 03.65.−w

Print publication: Issue 26 (4 July 2003)
Received 19 February 2003, in final form 25 April 2003
Published 18 June 2003