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Periodic Manifolds with Spectral Gaps

Olaf Post
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Abstract

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding Laplacian has at least $N$ open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.

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Olaf Post. 2002-07-14. Periodic Manifolds with Spectral Gaps. https://arxiv.org/abs/math-ph/0207017

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