Zeros of the wave functions of general polynomial potentials

Abstract

Analytic formulae for the moments of zeros of the wave function of the general polynomial potential V(x) = α MzM + αM–1zM–1 + αM–2zM–2 +...+ α1z are derived. Since the coefficients αM, α M–1,...α1 can be either real or complex constants, these formulae are valid for both Hermitian and non-Hermitian PT-symmetric (P — parity operator and T — time reversal operator) systems. These analytic formulae for the moments can be used to obtain polynomials, the roots of which are the zeros of the corresponding wave functions. The formulae derived here contain parameters of the potential αM, αM–1, ...α1 explicitly and locations of zeros can be calculated very efficiently with them.PACS Nos.: 3.65.–w, 3.65.Ge, 3.65.Sq, 4.25.–g