Extended continued fractions and energies of the anharmonic oscillators

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Abstract
We describe the analytic solution to the Schrödinger eigenvalue problem for the class of the central potentials V(r)=∑δ∈Zaδrδ, where a−2>−1/4, amax δ >0, Z is an arbitrary finite set of the integer or rational exponents, −2≤δ1<δ2<⋅⋅⋅<δI, and the couplings aδ satisfy only one auxiliary (formal, ‘‘superconfinement’’) restriction of the type aδI−1 >0. The formalism is based on an analysis of the asymptotic behavior of the explicit regular solution ψ(r) and issues in the formulation of the ‘‘secular’’ equation 1/L1(E)=0 which determines the binding energies. The main result is the rigorous construction of L1(E) as a generalized (‘‘extended’’) and convergent continued fraction. The method cannot be applied to the aδI−1 <0 cases—this disproves the closely related Hill-determinant approach as conjectured recently by Singh et al. for the simplest potentials with Z={−2,2,4,6} and Z={−2,−1,1,2}.