N-dimensional Coulomb-Sturmians with noninteger quantum numbers

TL;DR

This paper extends Coulomb-Sturmian functions to N-dimensions with noninteger quantum numbers, generalizing their mathematical framework and clarifying their relation to Guseinov's Psi-alpha-ETOs.

Contribution

It introduces a generalized N-dimensional Coulomb-Sturmian framework allowing noninteger quantum numbers and clarifies the connection to Guseinov's orbitals.

Findings

Coulomb-Sturmian functions satisfy derived N-dimensional differential equations.

Guseinov's Psi-alpha-ETOs are identified as shifted-dimensional Coulomb-Sturmians.

The generalization broadens the applicability of Coulomb-Sturmian functions in quantum mechanics.

Abstract

Coulomb-Sturmian functions are complete, orthonormal, and include the full spectrum of continuum states. They are restricted to integer values of quantum numbers, as imposed by boundary and orthonormality conditions. Bagci-Hoggan exponential-type orbitals remove this restriction through a generalization to quantum number with fractional order. The differential equations for N-dimensional Bagci-Hoggan orbitals are derived. It is demonstrated that Coulomb-Sturmian functions satisfy a particular case of these equations. Additionally, Guseinov's Psi-alpha-ETOs are identified as N-dimensional Coulomb-Sturmians with a shifted dimensional parameter alpha, rather than representing an independent complete orthonormal sets of basis in a weighted Hilbert space.