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Singular Potentials in Quantum Mechanics and Ambiguity in the Self-Adjoint Hamiltonian
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and $V(x)=g/x^2$ 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.
Keywords: quantum mechanics; singular potential; self-adjointness; boundary condition. 
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     title = {Singular {Potentials} in {Quantum} {Mechanics} and {Ambiguity} in the {Self-Adjoint} {Hamiltonian}},
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Tamáas Fülöp. Singular Potentials in Quantum Mechanics and Ambiguity in the Self-Adjoint Hamiltonian. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a106/

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