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 $V(x)=g/x^2$ 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.

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/

References in corpus (5)

• Moebius Structure of the Spectral Space of Schroedinger Operators with Point Interaction
• Origin of the anomalies: the modified Heisenberg equation
• Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables
• Classical Aspects of Quantum Walls in One Dimension
• Quantum Abacus

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• Exact time-dependent states for throat quantized toroidal AdS black holes
• Singularity resolution by lattice shifts in discretised quantum mechanics
• Quasinormal modes and the analytical continuation of non-self-adjoint operators