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4:00 pm Wednesday, November 20, 2013 Stulken Lecture: Gaps Between Eigenvalues of Quasi-Periodic SchrÃ¶dinger Operators IVby Michael Goldstein (University of Toronto) in HB 227- This is the last talk in the four lectures series on the size of the splitting between eigenvalues of quasi-periodic Schrodinger equation. The lecture will start with a discussion of some important details arising in the elimination of the resonances. In particular, we will see the importance of the so called Wegner estimate. The latter refers to the total mass of phases for which one of the eigenvalues of the matrix falls into a given small interval. This estimate is one of the central pieces of the Anderson Localization theory in all possible settings. After that we will discuss how exactly Cramer's rule combined with the elimination of resonances and Wegner estimate leads to the estimates on the decay of the eigenvector components. We will discuss the analog of the Avalanche Principle expansion for the logarithm of the characteristic determinants and we will see how this gives an expansion of the eigenvector components. This latter expansion is the key to the estimate of the splitting between eigenvalues. The reason for that is that the expansion exhibits stability with respect to the spectral parameter unless another resonance kicks in. For that matter different eigenvalues must be split since otherwise the corresponding eigenvectors cannot be orthogonal. The stability bound gives the size of the split.
Submitted by damanik@rice.edu |