Pseudo-Gaussian quantum models

TL;DR

This paper introduces a new family of one-dimensional quantum models with potentials that behave like Gaussians at infinity but resemble harmonic oscillators near zero, with matrix elements derived via generating functionals.

Contribution

The paper presents novel quantum models with unique Gaussian-like potentials and provides a method to compute their Hamiltonian matrix elements using generating functionals.

Findings

New quantum models with Gaussian asymptotics

Matrix elements derived from generating functionals

Potential to explore new quantum behaviors

Abstract

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 o scillator when $x\to 0$. 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.