Math 2.0

Operator theory Ordinary and partial differential operators Problem

Ten martini problem

Builds on Almost Mathieu operatorSpectrum of an operator

Problem

For real numbers λ,α,θ\lambda, \alpha, \theta, let Hλ,α,θH_{\lambda,\alpha,\theta} be the almost Mathieu operator on ℓ2(ℤ)\ell^2(\Z). A Cantor set in ℝ\R is a nonempty compact set with no isolated points and with empty interior. Is the spectrum of Hλ,α,θH_{\lambda,\alpha,\theta} a Cantor set for every irrational α\alpha, every λ≠0\lambda \ne 0 and every θ\theta?

Remarks

The answer is yes. Avila and Jitomirskaya proved it in a paper titled "The Ten Martini Problem", published in the Annals of Mathematics in 2009. Before that, Bellissard and Simon showed that the spectrum is a Cantor set for a generic set of pairs (λ,α)(\lambda, \alpha). The name comes from Mark Kac, who offered ten martinis for a solution; Barry Simon gave the problem this name in his list of problems on almost periodic Schrödinger operators.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Almost Mathieu operator, example, no evidence levelAlmost Mathieu operat…exampleSpectrum of an operator, concept, no evidence levelSpectrum of an operat…conceptTen martini problem, problem, no evidence levelTen martini problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Almost Mathieu operator example
  2. Spectrum of an operator concept
  3. Ten martini problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Almost_Mathieu_operator
About this page
Tag3X4ZQ41PXG
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Kindproblem
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MSC47B36, 81Q10, 47A10
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AboutAlmost Mathieu operator
UsesAlmost Mathieu operator, Spectrum of an operator
Changed2 November 2026

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