Differences between Robin and Neumann eigenvalues

Abstract

Let $\Omega \subset {\mathbb{R}}^2$ be a bounded planar domain, with piecewise smooth boundary $\partial \Omega$. For $\sigma >0$, we consider the Robin boundary value problem \[ -Δf =λf, \qquad \frac{\partial f}{\partial n} + σf = 0 \mbox{ on } \partial Ω\] where $\frac{\partial f}{\partial n}$ is the derivative in the direction of the outward pointing normal to $\partial \Omega$. Let $0<{\lambda }_0^{\sigma }\le {\lambda }_1^{\sigma }\le \dots$ 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 $2\mathrm{length}(\partial \Omega )/\mathrm{area}(\Omega )\cdot \sigma$ and in the smooth case, give an upper bound of $d_n(\sigma )\le C(\Omega )n^{1/3}\sigma$ 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.

arXiv commentSeveral changes. Added references and comments about higher dimensions and variable Robin function

References in corpus (6)

• Semiclassical Quantization of Billiards with Mixed Boundary Conditions
• Spectrum of a non-selfadjoint quantum star graph
• Singularities of the wave trace near cluster points of the length spectrum
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Cited by in corpus (4)

• Comparing the spectrum of Schrödinger operators on quantum graphs
• The Robin problem on rectangles
• Differences between Robin and Neumann eigenvalues on metric graphs
• On the Robin spectrum for the equilateral triangle