Extended continued fractions and energies of the anharmonic oscillators

Journal of Mathematical Physics - Tập 24 Số 5 - Trang 1136-1141 - 1983
Miloslav Znojil1
1Nuclear Physics Institute, Czechoslovak Academy of Sciences, 250 68 Řež near Prague, Czechoslovakia

Tóm tắt

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}.

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Journal of Mathematical Physics

Tập 24 Số 5

1136-1141

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