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Monodromy-free Schrodinger equations and Painlev\'e transcendents

Friday 22nd September 2006 - 15:30 to 16:30
INI Seminar Room 1

A Schroedinger operator with meromorphic potential is called monodromy-free if all solutions of the corresponding Schroedinger equation are meromorphic for all values of energy (so the corresponding monodromy in the complex plane is trivial). A nice class of examples is given by the so-called "finite-gap" operators, but in general the description of all monodromy-free operators is open even in the class of rational potentials, although in some special cases the answer is known (Duistermaat-Grunbaum, Gesztesy-Weikard, Oblomkov).

In the talk I will describe a class of Schroedinger operators with trivial monodromy, constructed in terms of the Painleve-IV transcendents and their higher analogues determined by the periodic dressing chains. We will discuss also a new interpretation and a fundamental role of the Stieltjes relations in this problem.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons