Pseudo-Gaussian quantum models

A new family of one-dimensional quantum models is proposed in terms of new potentials with a Gaussian asymptotic behavior but approaching to the potential of the harmonic oscillator when x → 0. It is shown that, in the energy basis of the harmonic oscillator, the matrix elements of the Hamiltonian operators of these new models can be derived from generating functionals. Pacs 03.65.Ge The progress in algebraic and numerical computation offers one new possibilities of analyzing new classical or quantum systems that can not be analytically solved. The Schrödinger equation with an attractive radial Gaussian potential was analyzed with different perturbation methods leading to numerical results of convenient accuracy [1]. Other quantum models with Gaussian barriers were considered in the theory of electronic transport in mesoscopic systems [2, 3]. Some time ago we studied a simple one-dimensional relativistic model of Gaussian well using generating functionals for finding the matrix elements of the Klein-Gordon operator in the energy basis of the non-relativistic harmonic oscillator (HO) [4]. In this framework computational methods allowed us to point out the structure of the discrete energy spectra of the Gaussian wells which indicates that inside the well the quantum motion is close to an harmonic one. Here we would like to continue this study in the case of the non-relativistic quantum system, looking for general one-dimensional models with pseudoGaussian potentials. Our purpose is to define a large family of models whose