## Abstract

Let Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary ∂ Ω . For σ > 0 , we consider the Robin boundary value problem - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where ∂ f ∂ n is the derivative in the direction of the outward pointing normal to ∂ Ω . Let 0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps d n ( σ ) : = λ n σ - λ n 0 . For a wide class of planar domains we show that there is a limiting mean value, equal to 2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Original language | English |
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Pages (from-to) | 1603-1635 |

Number of pages | 33 |

Journal | Communications in Mathematical Physics |

Volume | 388 |

Issue number | 3 |

DOIs | |

State | Published - Dec 2021 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s).

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics