2007/08/31 by Tamás Fülöp
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #quant-ph
paper · pdf · doi:10.3842/sigma.2007.107
published as SIGMA 3 (2007), 107, 12 pages · 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/
arxiv created 2007/11/16 · openalex publication_date 2007/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Schrdinger 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 selfadjoint 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.