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Y. C. Cantelaube

Publications and source records attributed to Y. C. Cantelaube.

2 recordsLinked to original sources
arXiv2012

Energy levels and extension of the Schrodinger operator

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.

quant-ph↗
arXiv2012

Solutions of the Schrödinger equation, boundary condition at the origin, and theory of distributions

In a central potential the usual resolution of the Schrödinger equation in spherical coordinates consists in determining the solutions R(r) or u(r) of the radial equations considered as the radial parts of the Schrödinger equation. However, the solutions must be supplemented with the boundary condition u(0) = 0 in order to rule out singular solutions. There is still no consensus to justify this condition, with good reason. It is based on a misunderstanding that comes from the fact that the radial equation in terms of R(r) is derived from the Schrödinger equation, and the radial equation in terms of u(r) from the former, by taking the Laplacians in the sense of the functions. By taking these Laplacians in the sense of the distributions, as it is required, we show that the radial equations are derived from the Schrodinger equation when their solutions are regular, but not when they are singular, so that the equations need not be supplemented with any supplementary condition such as u(0) = 0.

math-ph↗