Singular Potentials in Quantum Mechanics and Ambiguity in the Self-Adjoint Hamiltonian
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.
- Publication:
-
SIGMA
- Pub Date:
- November 2007
- DOI:
- 10.3842/SIGMA.2007.107
- arXiv:
- arXiv:0708.0866
- Bibcode:
- 2007SIGMA...3..107F
- Keywords:
-
- quantum mechanics;
- singular potential;
- self-adjointness;
- boundary condition;
- Quantum Physics
- E-Print:
- This is a contribution to the Proc. of the 3-rd Microconference "Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/