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

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- by Christopher K. R. T. Jones, Yuri Latushkin and Selim Sukhtaiev PDF
- Proc. Amer. Math. Soc.
**145**(2017), 363-377 Request permission

## 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 $\theta$.## References

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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
- MR Author ID: 95400
- ORCID: 0000-0002-2700-6096
- Email: ckrtj@email.unc.edu
**Yuri Latushkin**- Affiliation: Department of Mathematics, The University of Missouri, Columbia, Missouri 65211
- MR Author ID: 213557
- Email: latushkiny@missouri.edu
**Selim Sukhtaiev**- Affiliation: Department of Mathematics, The University of Missouri, Columbia, Missouri 65211
- Email: sswfd@mail.missouri.edu
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3565387