### Abstract

The accidental crossing of energy levels is studied for a number of exactly solvable PT-symmetric potentials in one spatial dimension. This phenomenon occurs when the potential possesses two series of bound-state levels discriminated by the q=± quasi-parity quantum number and a potential parameter is tuned to specific values. In contrast with the coalescing of two such real-energy levels with the same n quantum number and continuing as a complex conjugate pair, corresponding to the breakdown of PT symmetry, accidental crossing occurs for energy levels with different n and q. In this case the energy eigenvalues become degenerate, and the corresponding wave functions become linearly dependent. It is shown that besides the known examples, the PT-symmetric harmonic oscillator, Coulomb and Scarf II potentials, this phenomenon occurs for any member of the Natanzon potential class for which the q quantum number can be defined. Two such potentials are discussed as concrete examples: the PT-symmetric generalized Ginocchio potential and a four-parameter subset of the Natanzon potential class. These potentials have been described in detail previously, however, the accidental crossing of their energy eigenvalues has not been noticed then.

Original language | English |
---|---|

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Annals of Physics |

Volume | 380 |

DOIs | |

Publication status | Published - May 1 2017 |

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### Keywords

- Bound states
- Energy spectrum
- Exactly solvable potentials
- PT symmetry

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Accidental crossing of energy eigenvalues in PT-symmetric Natanzon-class potentials.** / Lévai, G.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Accidental crossing of energy eigenvalues in PT-symmetric Natanzon-class potentials

AU - Lévai, G.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - The accidental crossing of energy levels is studied for a number of exactly solvable PT-symmetric potentials in one spatial dimension. This phenomenon occurs when the potential possesses two series of bound-state levels discriminated by the q=± quasi-parity quantum number and a potential parameter is tuned to specific values. In contrast with the coalescing of two such real-energy levels with the same n quantum number and continuing as a complex conjugate pair, corresponding to the breakdown of PT symmetry, accidental crossing occurs for energy levels with different n and q. In this case the energy eigenvalues become degenerate, and the corresponding wave functions become linearly dependent. It is shown that besides the known examples, the PT-symmetric harmonic oscillator, Coulomb and Scarf II potentials, this phenomenon occurs for any member of the Natanzon potential class for which the q quantum number can be defined. Two such potentials are discussed as concrete examples: the PT-symmetric generalized Ginocchio potential and a four-parameter subset of the Natanzon potential class. These potentials have been described in detail previously, however, the accidental crossing of their energy eigenvalues has not been noticed then.

AB - The accidental crossing of energy levels is studied for a number of exactly solvable PT-symmetric potentials in one spatial dimension. This phenomenon occurs when the potential possesses two series of bound-state levels discriminated by the q=± quasi-parity quantum number and a potential parameter is tuned to specific values. In contrast with the coalescing of two such real-energy levels with the same n quantum number and continuing as a complex conjugate pair, corresponding to the breakdown of PT symmetry, accidental crossing occurs for energy levels with different n and q. In this case the energy eigenvalues become degenerate, and the corresponding wave functions become linearly dependent. It is shown that besides the known examples, the PT-symmetric harmonic oscillator, Coulomb and Scarf II potentials, this phenomenon occurs for any member of the Natanzon potential class for which the q quantum number can be defined. Two such potentials are discussed as concrete examples: the PT-symmetric generalized Ginocchio potential and a four-parameter subset of the Natanzon potential class. These potentials have been described in detail previously, however, the accidental crossing of their energy eigenvalues has not been noticed then.

KW - Bound states

KW - Energy spectrum

KW - Exactly solvable potentials

KW - PT symmetry

UR - http://www.scopus.com/inward/record.url?scp=85016307986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85016307986&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2017.03.001

DO - 10.1016/j.aop.2017.03.001

M3 - Article

AN - SCOPUS:85016307986

VL - 380

SP - 1

EP - 11

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

ER -