Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 107, 12 pages      arXiv:0708.0866      https://doi.org/10.3842/SIGMA.2007.107
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III

Singular Potentials in Quantum Mechanics and Ambiguity in the Self-Adjoint Hamiltonian

Tamás Fülöp
Montavid Research Group, Budapest, Soroksári út 38-40, 1095, Hungary

Received August 07, 2007, in final form November 08, 2007; Published online November 16, 2007

Abstract
For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and V(x) = g/x2 with the coefficient g in a certain range (x being a space coordinate in one or more dimensions), the corresponding Schrödinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The article discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behaviour of the wave functions around the singularity, and what physical differences emerge for different self-adjoint versions of the Hamiltonian. The paper reviews and interprets known results, with the intention to provide a practical guide for all those interested in how to approach these ambiguous situations.

Key words: quantum mechanics; singular potential; self-adjointness; boundary condition.

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