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Momentum Operators in Two Intervals: Spectra and Phase Transition
Journal article   Open access   Peer reviewed

Momentum Operators in Two Intervals: Spectra and Phase Transition

Palle Jorgensen, Steen Pedersen and Feng Tian
Complex Analysis and Operator Theory, Vol.7(6), pp.1735-1773
12/2013
DOI: 10.1007/s11785-012-0234-x
url
https://arxiv.org/pdf/1110.5948View
Open Access

Abstract

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.
Mathematics Special orthogonal functions Deficiency indices Symmetric and selfadjoint extensions Unbounded operators Fourier series 35F15 Spectral sets 42C10 Operator Theory Tilings Spectral pairs 47B25 47L60 Analysis 47A25 Mathematics, general Boundary values for linear first-order PDE

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