Composite Hermite and Anti-Hermite Polynomials

Abstract

The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed in a generalized form through introduction of a constant conjugation parameter  according to the transformation x x d d d d →  , where the conjugation parameter is set to unity (  = 1 ) at the end of the evaluations. Factorization in normal order form yields  -dependent composite eigenfunctions, Hermite polynomials and corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials and negative eigenvalues. The two sets of solutions are related by an  -sign reversal conjugation rule   → − . Setting  = 1 provides the standard Hermite polynomials and their partner antiHermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation