Pragmatic Self-adjoint Procedure in the Schrodinger Equation for the Inverse Square Potential

TL;DR

This paper explores a self-adjoint extension method for the Schrödinger equation with inverse square potentials, revealing how it introduces a bound state and modifies scattering amplitudes, ensuring mathematical consistency and physical relevance.

Contribution

It generalizes the pragmatic self-adjoint extension approach to include the inverse square potential, demonstrating the emergence of bound states and modified scattering amplitudes.

Findings

Single bound state appears after extension

Modified scattering amplitude includes an SAE-dependent factor

Bound state manifests as a pole in the scattering amplitude

Abstract

The self-adjoint extension (SAE) procedure is considered in the Schrodinger equation for potentials behaving as an attractive inverse square at the origin of coordinates. This approach guarantees self-adjointness of the radial Hamiltonian in three dimensions. It is shown that the single bound state appears after such an extension, which depends on SAE parameter. The same parameter arises for the scattering case as well, when the extension is made by orthogonality requirement. The closed form is derived for the modified scattering amplitude, which consists an extra factor depended on the SAE parameter. That guarantees the appearance of the same bound state in the form of the scattering amplitude pole. So, the generalization of pragmatic method is demonstrated in case of continuous spectrum.