arXiv subjects
Publications and source records attributed to Michael Semenov-Tian-Shansky.
Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, or alternatively by using the technique of the Quantum Inverse Scattering Method. A comparison of the two approaches, which is the purpose of the present review article, sheds a new light on Representation Theory and leads to a number of challenging questions.
We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdV-like nonlinear equations are discussed. The same approach is applied to second order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.