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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.
pdf (282 kb)
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