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1:00 pm Monday, November 11, 2013 Stulken Lecture: Gaps Between Eigenvalues of Quasi-Periodic SchrÃ¶dinger Operators IIIby Michael Goldstein (University of Toronto) in HB 453- This is the third talk in the four lectures series on the size of the splitting between eigenvalues of quasi-periodic Schrodinger equation. In this lecture we will discuss how to use the large deviations estimates for the analysis of eigenvectors of the matrix in question. We will see how the issue of resonances of the eigenvalues arises in this analysis. The latter can be described as an event when the characteristic polynomial of the problem evaluated at the same spectral parameter $E$ vanishes at some phase $\theta$ and also at a shifted phase $T^k(\theta)$. This issue naturally leads to analysis of the resultant of two polynomials in $E$,- one at the phase $\theta$ and another at the phase $T^k(\theta)$. That eliminates the resonant values of $E$. The local number of zeros of the characteristic polynomial is crucial for this analysis. The large deviation estimate is instrumental for that matter. Once the resonances are eliminated one obtains the exponential localization of the eigenvectors due to the Cramer's rule.
Submitted by damanik@rice.edu |