Mini course on Spectral Theory of Jacobi Operators and an Asymptotic Behavior of Orthogonal Polynomials
Spectral Methods in Mathematical Physics
06 March 10:00 - 12:00
Dimitri Yafaev - Université de Rennes 1; St Petersburg State University
Jacobi operators are discrete analogues of diﬀerential operators of Schrödinger type. Orthogonal polynomials Pn(z) can be regarded as eigenfunctions (perhaps, of the continuous spectrum) of Jacobi operators. So it looks quite natural to apply the well developed machinery of spectral theory of diﬀerential operators to the study of orthogonal polynomials. We are particularly interested in the asymptotic behavior of polynomials Pn(z) as n → ∞.
Here is a short plan of this course:
1. Diﬀerential equations: Weyl’s theory, short- and long-range perturbations, semiclassical Green-Liouville Ansatz.
2. Jacobi operators and their spectral measures. Direct and inverse spectral problems. Hilbert-Schmidt perturbations. Orthogonal polynomials as eigenfunctions of Jacobi operators.
3. The “free” Jacobi operator and Chebyshev polynomials. Short-range perturbations. Scattering theory, the perturbation determinant, the spectral shift function, the Szegö function. Example: point interaction. Generalizations of the Bernstein-Szegö asymptotic formulas.
4. Long-range perturbations. A semiclassical Ansatz for diﬀerence equations. Stationary scat-tering theory. Example: Pollaczek polynomials.
University of Zurich, UZH