Periodicity of the spectrum in dimension one

Details Periodicity of the spectrum in dimension one Periodicity of the spectrum in dimension one

2011-08-29

arxiv.org/abs/1108.5689v2

Mathematics

Classical Analysis and ODEs

Correction of an error pointed out by Dorin Dutkay; Lemma 1 now has a new proof

Scientific paper

A bounded measurable set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers ("frequencies") such that the exponential functions $e_\lambda(x) = \exp(2\pi i \lambda x)$, $\lambda\in\Lambda$, form a complete orthonormal system of $L^2(\Omega)$. Such a set $\Lambda$ is called a {\em spectrum} of $\Omega$. In this note we prove that any spectrum $\Lambda$ of a bounded measurable set $\Omega\subseteq\RR$ must be periodic.

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