Eur. Phys. J. B 31, 247–248 (2003) DOI: 10.1140/epjb/e2003-00028-4 T HE EUROPEAN P HYSICAL JOURNAL B A self consistent technique for singular potentials S. Datta a and J.K. Bhattacharjee Department of Theoretical Physics, Indian Association for the Cultivation of Science Jadavpur, Calcutta 700 032, India Received 29 August 2002 / Received in final form 20 November 2002 Published online 4 February 2003 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2003 Abstract. A technique used by Edwards and Singh [8] in the study of polymers is used to provide a simple analytic procedure for calculating the upper bound of the energies of a class of singular potentials. The accuracy of our procedure turns out to be better than 1%. PACS. 01.55.+b General physics – 02.90.+p Other topics in mathematical methods in physics Attractive potentials with a strong repulsive core are common in nuclear physics and molecular physics. A class of such potentials that have been studied ex- tensively [1–6,9] in the last two decades is the class represented by V (r)= r 2 + λ r α . Of particular interest is the ground state energy of a particle in this potential. In three dimensional space, the solution of Hψ = Eψ for the l = 0 state reduces to - d 2 u dr 2 +  r 2 + λ r α  u = Eu (1) where ψ = u/r and the boundary conditions on u is that it vanishes at r = 0. For α ≥ 3, these potentials exibit the so called Klauder phenomenon [4] where perturbation theory has to be ordered in fractional powers of λ, rather than integer powers for λ ≪ 1. For α = 2, the model is exactly solvable and as one raises α beyond 2, the difficulty that is going to set in at α = 3 makes its presence felt by making convergence of numerical procedures slow. Very recently Hall and Saad [7] have adopted a clever variational scheme that yields accurate answers with much less labour. In this note we adopt an old argument of Edwards and Singh [8], set up to deal with a situation in polymers, to obtain a simple analytic formula for getting the upper bound of the ground state energy which is within 1% of the accurate variational and numerical results of Hall and Saad [7] for all λ. Specifically, we compare with Hall and Saad over a range of 6 decades in λ for α =2.5. The method of Edwards and Singh [8] exploits a neighboring exact result and uses selfconsistency. To implement it, we write, H = - d 2 dr 2 + r 2 + λβ〈 1 r α-2 〉 1 r 2 + λ r α - λβ〈 1 r α-2 〉 1 r 2 (2) a e-mail: sumita@bose.res.in = - d 2 dr 2 + r 2 + A r 2 + λ  1 r α - β〈 1 r α-2 〉 1 r 2  = H 0 + λH ′ (3) where A = λβ〈 1 r α-2 〉 (4) H 0 = - d 2 dr 2 + r 2 + A r 2 (5) H ′ = λ ×  1 r α - β〈 1 r α-2 〉 1 r 2  . (6) The eigenvalues and eigenfunctions of H 0 are exactly known. We first choose β by requiring that the first or- der perturbation correction to eigenvalues of H 0 vanish. This implies (we will always focus on the ground state) that 〈H ′ 〉 = 0 and hence β =  〈 1 r α-2 〉〈 1 r 2 〉  -1 〈 1 r α 〉 (7) where all averages are taken with respect to the ground state of H 0 . With this choice of β, A = λ〈 1 r α 〉/〈 1 r 2 〉. (8) The ground state energy is given by (using the known spectrum) of H 0 , E =2+ √ 1+4A (9)