Approximate energy expressions for confining polynomial potentials

Abstract

tly developed asymptotic energy expansion (AEE) method is applied to obtain totic energy expansions (AEEs) of general polynomial potentials. These sions contain coefficients of the polynomial potentials explicitly. The asymptotic sions produce very accurate eigen energies. Recurrence relations derived here can d to obtain asymptotic expansions of polynomial potentials of any degree. Energy value expressions are presented for the 4th, 6th, 8th and 10th degree polynomial ials. The expansions obtained with AEE method for polynomial potentials have blance with WKB expressions obtained by Bender et al for the potentials ( even). N x = N UCTION re have been great interest in anharmonic oscillator potentials due to their in quantum field theory, nuclear physics, particle physics, solid-state d atomic and molecular physics. Both perturbative and non-perturbative have been used in the literature to study these potentials with varying this paper, first, we study general polynomial potentials of even degree ly developed AEE method and then explicit expansions are derived for sixth, eighth, and tenth degree polynomial potentials. The asymptotic eigen ansions are written as h ) 2 1 ( + = n g author (E-mail: asiri@ifs.ac.lk) A. Nanayakkara and V. Bandara /Sri Lankan Journal of Physics, Vol.3 (2002) 17-37 where is the k n E α k α th power of the eigen energy, corresponding to nth excited state and coefficients are known explicitly as polynomials in coefficients of the polynomial potential. The AEE method is well suited for polynomial potentials because they have the property s bk '