Preface

The application of group theoretic methods to quantum mechanics has a long

and distinguished history, dating back to the early investigations ofWeyl, Wigner

and Bargmann. In the "standard" approach, symmetries of the classical physical

problem become symmetries (commuting operators) of the associated quantum

mechanical Schrodinger operator. In favorable situations, this implies that the

bound states of the associated stationary Schrodinger equation decompose into

irreducible representations for the symmetry group, and the powerful theoretical

methods from representation theory can be effectively used to analyze such prob-

lems; vice versa, the applications to quantum mechanics have themselves served

as a powerful catalyst and motivation for the enormous amount of mathematical

research effort that has gone into the study of representation theory.

The present volume

is

devoted to a range of important new directions and

ideas concerning the applications of Lie groups and Lie algebras to Schrodinger

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; nevertheless, the representation theory can still be effectively

employed to analyze at least part of the spectrum of the operator. Such methods

have their origin in the work of Dothan, Gell-Mann, Ne'eman, Barut and Bohm

in the mid 1960's. The concept of a spectrum-generating algebra was introduced

in nuclear physics, molecular physics, and scattering theory by Iachello, Arima,

and Levine, in the 1970's and 80's, leading to the general concept of a Lie alge-

braic Schrodinger operator. In the mid 1980's, Turbiner, Shifman, Ushveridze,

and collaborators introduced the more restrictive notion of a "quasi-exactly solv-

able" Schrodinger operator, for which the hidden symmetry algebra leads to the

determination of a part of the spectrum by purely algebraic means. The synthe-

sis and extensions of these seminal ideas has proved to be of great fruitfulness,

not only in physical problems, but in the mathematical theory required to un-

derstand and classify the original examples.

In light of the rapidly developing subject, we decided that the time was ripe

for bringing together, perhaps for the first time, a group of mathematicians and

physicists working in areas closely related to these themes. As you can see from

the contributions contained herein, a wide variety of physical applications and

vii

The application of group theoretic methods to quantum mechanics has a long

and distinguished history, dating back to the early investigations ofWeyl, Wigner

and Bargmann. In the "standard" approach, symmetries of the classical physical

problem become symmetries (commuting operators) of the associated quantum

mechanical Schrodinger operator. In favorable situations, this implies that the

bound states of the associated stationary Schrodinger equation decompose into

irreducible representations for the symmetry group, and the powerful theoretical

methods from representation theory can be effectively used to analyze such prob-

lems; vice versa, the applications to quantum mechanics have themselves served

as a powerful catalyst and motivation for the enormous amount of mathematical

research effort that has gone into the study of representation theory.

The present volume

is

devoted to a range of important new directions and

ideas concerning the applications of Lie groups and Lie algebras to Schrodinger

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; nevertheless, the representation theory can still be effectively

employed to analyze at least part of the spectrum of the operator. Such methods

have their origin in the work of Dothan, Gell-Mann, Ne'eman, Barut and Bohm

in the mid 1960's. The concept of a spectrum-generating algebra was introduced

in nuclear physics, molecular physics, and scattering theory by Iachello, Arima,

and Levine, in the 1970's and 80's, leading to the general concept of a Lie alge-

braic Schrodinger operator. In the mid 1980's, Turbiner, Shifman, Ushveridze,

and collaborators introduced the more restrictive notion of a "quasi-exactly solv-

able" Schrodinger operator, for which the hidden symmetry algebra leads to the

determination of a part of the spectrum by purely algebraic means. The synthe-

sis and extensions of these seminal ideas has proved to be of great fruitfulness,

not only in physical problems, but in the mathematical theory required to un-

derstand and classify the original examples.

In light of the rapidly developing subject, we decided that the time was ripe

for bringing together, perhaps for the first time, a group of mathematicians and

physicists working in areas closely related to these themes. As you can see from

the contributions contained herein, a wide variety of physical applications and

vii