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arXiv:2406.15530

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

Anzor Khelashvili, Teimuraz Nadareishvili

Published Jun 21, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2406.15530

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The quantum mechanics of inverse square potentials in one dimension is usually studied through renormalization, self-adjoint extension and WKB approximation. This paper shows that such potentials may be investigated within the framework of the position-dependent mass quantum mechanics formalism under the usual boundary conditions. As a result, exact discrete bound state solutions are expressed in terms of associated Laguerre polynomials with negative energy spectrum using the Nikiforov-Uvarov method for the repulsive inverse square potential.
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