Integrable Systems and Riemann Surfaces Lecture Notes (preliminary version)
نویسنده
چکیده
1 KdV equation and Schrödinger operator 2 1.1 Integrability of Korteweg – de Vries equation . . . . . . . . . . . . . . . . . . 2 1.2 Elements of scattering theory for the Schrödinger operator . . . . . . . . . . . 5 1.3 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dressing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Particular case: reflectionless potential . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Bloch spectrum of the Schrödinger operator with a periodic potential . . . . . 20 1.7 Properties of the monodromy matrix . . . . . . . . . . . . . . . . . . . . . . . 25 1.8 Differentiating with respect to the spectral parameter . . . . . . . . . . . . . 26 1.9 Finite gap case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.10 The theory of KdV hierarchy-1. Recursion relations and generating functions. 31 1.11 Stationary equations of KdV hierarchy, spectral curves and egenfunctions of the Schrödinger operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
منابع مشابه
Intersection Theory , Integrable Hierarchies and Topological Field Theory ∗
In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevich naturally appear as τ -functions of integrable hierarchies related to topological minimal model...
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