This volume is devoted to a range of important new ideas arising in the applications of Lie groups and Lie algebras to Schrödinger operators and associated quantum mechanical systems. In these applications, the group does not appear as a standard symmetry group, but rather as a "hidden" symmetry group whose representation theory can still be employed to analyze at least part of the spectrum of the operator. In light of the rapid developments in this subject, a Special Session was organized at the AMS meeting at Southwest Missouri State University in March 1992 in order to bring together, perhaps for the first time, mathematicians and physicists working in closely related areas. The contributions to this volume cover Lie group methods, Lie algebras and Lie algebra cohomology, representation theory, orthogonal polynomials, \(q\)-series, conformal field theory, quantum groups, scattering theory, classical invariant theory, and other topics. This volume, which contains a good balance of research and survey papers, presents a look at some of the current developments in this extraordinarily rich and vibrant area.

Readership

Pure mathematicians, applied mathematicians, and theoretical physicists.

Table of Contents

• B. Abraham-Shrauner and A. Guo -- Hidden symmetries of differential equations
• Y. Alhassid -- Algebraic methods in scattering
• C. M. Bender -- Exact solutions to operator differential equations
• L. C. Biedenharn -- The algebra of tensor operators for the unitary groups
• P. Feinsilver -- Lie groups and probability
• D. Flath -- Coherent tensor operators
• R. Floreanini and L. Vinet -- \({\scr U}_q(sl(2))\) and q -special functions
• J. N. Ginocchio -- The group representation matrix in quantum mechanical scattering
• A. González-López, N. Kamran, and P. J. Olver -- Quasi-exact solvability
• P. E. Jorgensen -- Quantization and deformation of Lie algebras
• F. Iachello -- Algebraic theory
• D. J. Kaup -- The time-dependent Schrödinger equation in multidimensional integrable evolution equations
• E. G. Kalnins, W. Miller, Jr., and S. Mukherjee -- Models of \(q\)-algebra representations: Matrix elements of \(U_q(su_2)\)
• J. Paldus -- Many-electron correlation problem and Lie algebras
• M. A. Shifman -- Quasi-exactly-solvable spectral problems and conformal field theory
• A. Turbiner -- Lie-algebras and linear operators with invariant subspaces