## Abstract

Using WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values

for which the spectrum of the quartic anharmonic oscillator in the complex plane

with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob’ev–Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials.

for which the spectrum of the quartic anharmonic oscillator in the complex plane

with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob’ev–Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials.

Original language | English |
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Article number | 52 |

Number of pages | 62 |

Journal | Communications in Mathematical Physics |

Volume | 405 |

Issue number | 2 |

DOIs | |

Publication status | Published - 20 Feb 2024 |

### Bibliographical note

Publisher Copyright:© The Author(s) 2024.

## Keywords

- Anharmonic oscillator
- WKB expansion
- degenerate orthogonal polynomials
- Painleve' II equation