Schroedinger semigroups for vector fields

Journal Article · Fri Mar 01 04:00:00 EST 1985 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.526631· OSTI ID:6105648

Osborn, T A; Corns, R A; Fujiwara, Y

Suppose H is the Hamiltonian that generates time evolution in an N-body, spin-dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r x r matrix operator given by the sum of the negative Laplacian -..delta../sub x/ on the (d = 3N)-dimensional Euclidean space R/sup d/ plus a Hermitian local matrix potential W(x). Uniform higher-order asymptotic expansions are derived for the time-evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix-valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex-valued matrix measures. This class is suitable for the N-body problem since interactions of this type do not necessarily decrease as Vertical BarxVertical Bar..-->..infinity. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup )e/sup -z/HVertical BarRe z>0).

• Research Organization: Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742
• OSTI ID: 6105648
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 26:3; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

645400* -- High Energy Physics-- Field Theory
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ANGULAR MOMENTUM
EUCLIDEAN SPACE
FOURIER TRANSFORMATION
GROUP THEORY
HAMILTONIANS
INTEGRAL TRANSFORMATIONS
KERNELS
MANY-BODY PROBLEM
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATHEMATICS
MATRICES
PARTICLE PROPERTIES
PARTICLES
POTENTIALS
QUANTUM OPERATORS
RIEMANN SPACE
SERIES EXPANSION
SPACE
SPIN
TENSORS
TRANSFORMATIONS
VECTOR FIELDS
VECTORS