Energy levels and extension of the Schrodinger operator

TL;DR

This paper introduces an extension of the Schrödinger operator that accounts for energy levels associated with singular eigenfunctions, providing a more comprehensive framework for understanding vibrational levels in quantum systems.

Contribution

It defines a new operator extension of the Schrödinger operator that includes singular eigenfunctions, broadening the understanding of energy levels beyond the traditional Hamiltonian.

Findings

The extended operator accounts for energy levels with singular eigenfunctions.

Energy levels are given by eigenfunctions of this extended operator, not just the Hamiltonian.

The extension involves embedding in distributions of R3, capturing singular solutions.

Abstract

Although energy levels are often given by solutions of the radial equation such that u(0) is non zero, and hence by first-order singular functions which are not eigenfunctions of H, the latter is always considered as the only operator that gives energy levels. Vibrational levels of diatomic molecules are a usual example. We show that the operator which has singular eigenfunctions, or pseudofunctions, that give energy levels, is the operator whose action on pseudofunctions amounts to the embedding in the distributions of R3 of their Hamiltonian in R3/{0}. When its eigenfunctions are regular, this operator amounts to H. Energy levels, which are given by eigenfunctions of H when u(0) is zero, are thus given in any case by eigenfunctions of this operator, which is an extension of the Schrodinger operator, but not of the Hamiltonian.