Explicit energy expansion for general odd degree polynomial potentials

TL;DR

This paper presents an explicit analytic formula for the asymptotic eigenenergy expansion of odd degree polynomial potentials, enabling efficient semiclassical calculations for non-Hermitian systems with complex parameters.

Contribution

It derives a nearly explicit formula for eigenenergy expansions of odd degree polynomial potentials, including complex cases, using contour integrations in the complex plane.

Findings

Accurate eigenenergy approximations for real and complex spectra.

Efficient semiclassical expressions up to any order.

Applicable to non-Hermitian polynomial potentials.

Abstract

In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form $V(x)=(ix)^{2N+1}+\beta _{1}x^{2N}+\beta _{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +\beta _{2N}x$ where $\beta _{k}^{\prime }$s are real or complex for $1\leq k\leq 2N$. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters $\beta _{1},\beta _{2}....$ and $\beta _{2N}$ of the potential. Unlike in the even degree polynomial case, the highest order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex turning points which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.