Spectral theory of multiple intervals
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- by Palle Jorgensen, Steen Pedersen and Feng Tian PDF
- Trans. Amer. Math. Soc. 367 (2015), 1671-1735 Request permission
Abstract:
We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space). The models allow for a scale of complexity, indexed by the natural numbers $\mathbb {N}$. For each $n\in \mathbb {N}$, we get families of selfadjoint operators indexed by: (i) the unitary matrix group $U(n)$, and by (ii) a prescribed set of $n$ non-overlapping intervals. Take $\Omega$ to be the complement in $\mathbb {R}$ of $n$ fixed closed finite and disjoint intervals, and let $L^{2}(\Omega )$ be the corresponding Hilbert space. Moreover, given $B\in U(n)$, then both the lengths of the respective intervals, and the gaps between them, show up as spectral parameters in our corresponding spectral resolutions within $L^{2}(\Omega )$. Our models have two advantages. One, they encompass realistic features from quantum theory, from acoustic wave equations and their obstacle scattering, as well as from harmonic analysis.
Secondly, each choice of the parameters in our models, $n\in \mathbb {N}$, $B\in U(n)$, and interval configuration, allows for explicit computations, and even for closed-form formulas: Computation of spectral resolutions, of generalized eigenfunctions in $L^{2}(\Omega )$ for the continuous part of the spectrum, and for scattering coefficients. Our models further allow us to identify embedded point-spectrum (in the continuum), corresponding, for example, to bound-states in scattering, to trapped states, and to barriers in quantum scattering. The possibilities for the discrete atomic part of the spectrum includes both periodic and non-periodic distributions.
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Additional Information
- Palle Jorgensen
- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 95800
- ORCID: 0000-0003-2681-5753
- Email: jorgen@math.uiowa.edu
- Steen Pedersen
- Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435
- MR Author ID: 247731
- Email: steen@math.wright.edu
- Feng Tian
- Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435
- Email: feng.tian@wright.edu
- Received by editor(s): February 2, 2012
- Received by editor(s) in revised form: January 20, 2013
- Published electronically: October 1, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 1671-1735
- MSC (2010): Primary 47L60, 47A25, 47B25, 35F15, 42C10, 34L25, 35Q40, 81Q35, 81U35, 46L45, 46F12
- DOI: https://doi.org/10.1090/S0002-9947-2014-06296-X
- MathSciNet review: 3286496
Dedicated: Dedicated to the memory of William B. Arveson