Spectral Analysis for the Generalized Hermite Polynomials

Abstract

The operator theory associated with the Hermite polynomials does not extend to the generalized Hermite polynomials because the even and odd polynomials satisfy different differential equations. We show that this leads to two problems, each of interest on its own. We then weld them together to form a united spectral expansion. In addition, the exponent $\mu$ in the weight $|x{|^{2\mu }}{e^{ - {x^2}}}$ has traditionally always been greater than $- \frac {1}{2}$. We show what happens if $\mu \leq - \frac {1}{2}$. Finally, we examine the differential equations in left-definite spaces.