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Spectral theory of multiple intervals


Authors: Palle Jorgensen, Steen Pedersen and Feng Tian
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
Published electronically: October 1, 2014
MathSciNet review: 3286496
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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
Email: jorgen@math.uiowa.edu

Steen Pedersen
Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435
Email: steen@math.wright.edu

Feng Tian
Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435
Email: feng.tian@wright.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06296-X
Keywords: Unbounded operators, deficiency indices, Hilbert space, reproducing kernels, boundary values, unitary one-parameter group, generalized eigenfunctions, direct integral, multiplicity, scattering theory, obstacle scattering, quantum states, quantum-tunneling, Lax-Phillips, exterior domain, translation representation, spectral representation, spectral transforms, scattering operator, Poisson kernel, $SU(n)$, Dirac comb, exponential polynomials, unitary dilation, contraction semigroup, Shannon kernel, continuous and discrete spectrum, scattering poles
Received by editor(s): February 2, 2012
Received by editor(s) in revised form: January 20, 2013
Published electronically: October 1, 2014
Dedicated: Dedicated to the memory of William B. Arveson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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