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Finite gap Jacobi matrices: Szegö's theorem and Szegö asymptoticsMathematical Physics & Probability
|Speaker: ||Jacob Christiansen, Caltech|
|Location: ||2112 MSB|
|Start time: ||Thu, Mar 20 2008, 12:00AM|
In the spectral theory of orthogonal polynomials (or Jacobi matrices) the
goal is to relate information about the recurrence coefficients (=Jacobi
parameters) to information about the measure of orthogonality (=spectral
measure). When the essential spectrum is an interval, the situation is well
understood. But what happens when gaps occur?
In the talk I'll discuss results for Jacobi matrices whose essential
spectrum is a finite union of closed intervals. In particular, I'll show how
to obtain Szegö's theorem from a non-local step-by-step sum rule. Moreover,
I'll explain how one can use results of Remling and Sodin-Yuditskii to prove
that the Jacobi parameters approach a single point on the isospectral torus
in the setting of Szegö's theorem.
The talk is based on joint work with Barry Simon and Maxim Zinchenko. Our
results rely on potential theory and analytic function theory, with an
important link to Riemann surfaces and Fuchsian groups.