Jeffrey J. Mitchell

Coherent states on spheres

We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere Sd. The coherent states are labeled by points in the associated phase space T*(Sd). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of Hall and Stenzel, we give here a substantially different description based on ideas of Thiemann and of Kowalski and Rembielinski. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.

Coherent states for a 2-sphere with a magnetic field

We consider a particle moving on a 2-sphere in the presence of a constant magnetic field. Building on our earlier work in the nonmagnetic case we construct coherent states for this system. The coherent states are labeled by points in the associated phase space, the (co)tangent bundle of S2. They are constructed as eigenvectors for certain annihilation operators and expressed in terms of a certain heat kernel. These coherent states are not of Perelomov type but rather are constructed according to the ?complexifier? approach of Thiemann. We describe the Segal?Bargmann representation associated with the coherent states which is equivalent to a resolution of the identity.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Coherent states: mathematical and physical aspects?.

Isometry theorem for the Segal–Bargmann transform on a noncompact symmetric space of the complex type

We consider the Segal–Bargmann transform on a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the dual compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

AbstractLet G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let ρt be the heat kernel on G; that is, ρt is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from ρt and its derivatives, are the compact group analogs of the classical Hermite polynomials on

The Segal-Bargmann transform for noncompact symmetric spaces of the complex type

\mathbb{R}^n

SHORT TIME BEHAVIOR OF HERMITE FUNCTIONS ON COMPACT LIE GROUPS

. Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on

THE SEGAL-BARGMANN TRANSFORM FOR COMPACT QUOTIENTS OF SYMMETRIC SPACES OF THE COMPLEX TYPE

\mathfrak{g}\;\; = \;\;Lie(G)