Counting spectrum via the Maslov index for one dimensional 𝜃-periodic Schrödinger operators

Jones, Christopher; Latushkin, Yuri; Sukhtaiev, Selim

doi:10.1090/proc/13192

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 $\theta$.

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