### 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 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.

Original language | English |
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Article number | 107 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 3 |

DOIs | |

Publication status | Published - 2007 |

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### Keywords

- Boundary condition
- Quantum mechanics
- Self-adjointness
- Singular potential

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology
- Mathematical Physics

### Cite this

**Singular potentials in quantum mechanics and ambiguity in the self-adjoint Hamiltonian.** / Fülöp, T.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Singular potentials in quantum mechanics and ambiguity in the self-adjoint Hamiltonian

AU - Fülöp, T.

PY - 2007

Y1 - 2007

N2 - 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 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.

AB - 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 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.

KW - Boundary condition

KW - Quantum mechanics

KW - Self-adjointness

KW - Singular potential

UR - http://www.scopus.com/inward/record.url?scp=84889234683&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84889234683&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2007.107

DO - 10.3842/SIGMA.2007.107

M3 - Article

AN - SCOPUS:84889234683

VL - 3

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

M1 - 107

ER -