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Counting spectrum via the Maslov index for one dimensional $ \theta-$periodic Schrödinger operators


Authors: Christopher K. R. T. Jones, Yuri Latushkin and Selim Sukhtaiev
Journal: Proc. Amer. Math. Soc. 145 (2017), 363-377
MSC (2010): Primary 53D12, 34L40; Secondary 37J25, 70H12
DOI: https://doi.org/10.1090/proc/13192
Published electronically: July 6, 2016
MathSciNet review: 3565387
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Abstract: We study the spectrum of the Schrödinger operators with $ n\times n$ matrix valued potentials on a finite interval subject to $ \theta -$periodic boundary conditions. For two such operators, corresponding to different values of $ \theta $, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to $ \theta $ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a fixed $ \theta $.


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Additional Information

Christopher K. R. T. Jones
Affiliation: Department of Mathematics, The University of North Carolina, Chapel Hill, North Carolia 27599
Email: ckrtj@email.unc.edu

Yuri Latushkin
Affiliation: Department of Mathematics, The University of Missouri, Columbia, Missouri 65211
Email: latushkiny@missouri.edu

Selim Sukhtaiev
Affiliation: Department of Mathematics, The University of Missouri, Columbia, Missouri 65211
Email: sswfd@mail.missouri.edu

DOI: https://doi.org/10.1090/proc/13192
Keywords: Schr\"odinger equation, Hamiltonian systems, eigenvalues, stability, differential operators, discrete spectrum
Received by editor(s): October 14, 2015
Received by editor(s) in revised form: March 9, 2016
Published electronically: July 6, 2016
Additional Notes: This work was supported by the NSF grant DMS-1067929, by the Research Board and Research Council of the University of Missouri, and by the Simons Foundation.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2016 American Mathematical Society