Hill's potentials in weighted Sobolev spaces and their spectral gaps

Details Hill's potentials in weighted Sobolev spaces and their spectral gaps Hill's potentials in weighted Sobolev spaces and their spectral gaps

2009-08-04

arxiv.org/abs/0908.0491v2

Mathematics

Spectral Theory

28 pages. Small cosmetic changes

Scientific paper

We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential of Hill's equation to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths belong to a similarly weighted sequence space. An extension of this result to complex potentials is proven as well. We also recover Trubowitz results about analytic potentials. The proof essentially employs the implicit function theorem.

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