Asymptotic behavior of eigenenergies of nonpolynomial oscillator potentials V(x) = x2N + (λx m1)/(1 + gx m2)

Abstract

Analytic semiclassical energy expansions of nonpolynomial oscillator (NPO) potentials V(x) = x2N + (λx[Formula: see text])/(1 + gx[Formula: see text]) are obtained for arbitrary positive integers N, m1, and m2, and the real parameters λ and g using the asymptotic energy expansion (AEE) method. Because the AEE method has been previously developed only for polynomial potentials, the method is extended with new types of recurrence relations. It is then applied to the preceding general NPO to obtain expressions for quantum action variable J in terms of E and the parameters of the potential. These expansions are power series in energy and the coefficients of the series contain parameters λ and g explicitly. To avoid the singularities in the potential we only consider the cases where both λ and g are non-negative at the same time. Using the AEE expressions, it is shown that, for certain classes of NPOs, if potentials have the same N, and the same m1 – m2 or m1 – 2m2 then they have the same asymptotic eigenspectra. It was also shown that for certain cases, both λ and –λ as well as g and –g will produce the same asymptotic energy spectra. Analytic expressions are also derived for asymptotic level spacings of general NPOs in terms of λ and g.