Momentum Operators in Two Intervals: Spectra and Phase Transition

Details Momentum Operators in Two Intervals: Spectra and Phase Transition Momentum Operators in Two Intervals: Spectra and Phase Transition

2011-10-26

arxiv.org/abs/1110.5948v1

Mathematics

Spectral Theory

35 pages, 20 figures

Scientific paper

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.

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