Asymptotic Energy Expansion for Rational Power Polynomial Potentials

Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form V (x) = x2n/m + b1xn1/m1 + b2xn2/m2 + ... + bNxnN/mN where n, m, n1, m1, ..., nN, mN are positive integers while coefficients bk ∈ ℂ. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of Γ functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.