Factorization procedure and new generalized Hermite functions Marco A. Reyes ∗ and M. Ranfer´ı Guti´errez Departamento de F´ısica, DCI Campus Le´ on, Universidad de Guanajuato Apdo. Postal E143, 37150 Le´ on,Gto., M´ exico We propose an alternative factorization for the simple harmonic oscillator hamiltonian which includes Mielnik’s isospectral factorization as a particular case. This factorization is realized in two non-mutually adjoint operators whose inverse product, in the simplest case, lead to a new Sturm-Liouville eigenvalue equation which includes Schr¨ odinger’s equation for the oscillator and Hermite’s equation as particular cases for limiting values of the factorization’s parameter, and whose eigenfunctions allow us to define new generalized Hermite functions. PACS numbers: 02.30.Gp, 02.30.Hq, 03.65.Fd Abstract arXiv:1002.1344v1 [math-ph] 6 Feb 2010 ∗ Corresponding author. Electronic mail: marco@fisica.ugto.mx 1