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Brian C. Hall

  1. Complex structures adapted to magnetic flows.

    Wed, 01/11/2012 - 15:01 — SomNambul
    Authors: Brian C. Hall, William D. Kirwin
    Subjects: Symplectic Geometry
    Abstract

    Let $M$ be a compact real-analytic manifold, equipped with a real-analytic
    Riemannian metric $g,$ and let $\beta$ be a closed real-analytic 2-form on $M$,
    interpreted as a magnetic field. Consider the Hamiltonian flow on $T^*M$ that
    describes a charged particle moving in the magnetic field $\beta$. Following an
    idea of T.

  2. Coherent states for a 2-sphere with a magnetic field.

    Thu, 12/08/2011 - 15:01 — SomNambul
    Authors: Brian C. Hall, Jeffrey J. Mitchell
    Subjects: Mathematical Physics
    Abstract

    We consider a particle moving on a 2-sphere in the presence of a constant
    magnetic field. Building on 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 S^2. 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 T.

  3. The Segal-Bargmann transform for compact quotients of symmetric spaces of the complex type.

    Thu, 11/05/2009 - 16:01 — SomNambul
    Authors: Brian C. Hall, Jeffrey J. Mitchell
    Subjects: Mathematical Physics
    Abstract

    Let G be a connected complex semisimple group, assumed to have trivial
    center, and let K be a maximal compact subgroup of G. Then G/K, with a fixed
    G-invariant Riemannian metric, is a Riemannian symmetric space of the complex
    type. Now let Gamma be a discrete subgroup of G that acts freely and
    cocompactly on G/K. We consider the Segal--Bargmann transform, defined in terms
    of the heat equation, on the compact quotient Gamma\G/K. We obtain isometry and
    inversion formulas precisely parallel to the results we obtained previously for
    globally symmetric spaces of the complex type.

  4. Toeplitz operators on generalized Bergman spaces.

    Wed, 08/19/2009 - 19:30 — SomNambul
    Authors: Kamthorn Chailuek, Brian C. Hall
    Subjects: Complex Variables
    Abstract

    We consider the weighted Bergman spaces HL^2(B^d,\mu_{\lambda}), where
    d\mu_\lambda(z)=c_{\lambda}(1-|z|^2)^lambda d\tau, \tau being the hyperbolic
    volume measure. These spaces are nonzero if and only if \lambda>d. For
    0<\lambda\leq d, spaces with the same formula for the reproducing kernel can be
    defined using a Sobolev-type norm. We define Toeplitz operators on these
    generalized Bergman spaces and investigate their properties.

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