Quantum Scattering Transformation

Abstract

In one-dimensional mathematical physics we know three successful lines of development: the Bethe — Hulthen theory of quantum magnetics, the Onsager — Baxter theory of planar models in statistical mechanics and the Gardner — Greene — Kruskal — Miura — Zakharov — Shabat inverse scattering method in classical field theory. Recently it has been realized that after quantization of the latter method, a unified approach to all three of these lines emerges. This new method — the quantum inverse method or quantum scattering transformation (QST) — is now three years old and continues to develop rapidly in several centers (Leningrad Steklov Institute, Fermilab, Batavia, Freiburg University, etc.). There exist several surveys of the basic ideas of QST^{1), 2), 3), 4)}. In this text I shall mention some very recent developments made in the Leningrad group. Needless to say, their authors helped me in compiling this review.