T. A. Osborn

Symplectic areas, quantization, and dynamics in electromagnetic fields

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: A choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

Time evolution kernels: uniform asymptotic expansions

For a wide class of self‐adjoint Schrodinger Hamiltonians, a detailed description of the time evolution kernel is obtained. In a setting of a d‐dimensional Euclidean space without boundaries, the Schrodinger Hamiltonian H is the sum of the negative Laplacian plus a real‐valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N‐body problem, since they do not necessarily decrease as ‖x‖→∞. An asymptotic expansion in the complex parameter z, around z=0, is derived for the family of kernels Uz(x,y) corresponding to the analytic semigroup {e−zH: Re z>0}, which is uniform in the coordinate variables x and y. The asymptotic expansion has a simple semiclassical interpretation. Furthermore, an explicit bound for the remainder term in the asymptotic expansion is found. The expansion and the remainder term bound continue to the time axis boundary z=it/...

A phase‐space technique for the perturbation expansion of Schrödinger propagators

A perturbation theory for Schrodinger and heat equations that is based on phase‐space variables is developed. The Dyson series representing the evolution kernel is described in terms of two basic classical quantities: the free classical motion along flat space geodesics and the Green function for the Jacobi operator in phase space. Further, for problems with Abelian interactions it is demonstrated that the perturbation theory may be summed to all orders yielding an exponentiated connected graph description for the evolution kernel. Connected graph representations provide an efficient method of constructing various semiclassical approximations wherein expansion coefficients are directly determined by explicit cluster integrals. This type of application is discussed for the case of Schrodinger and heat equations with external electromagnetic fields. Detailed expressions for coefficients are obtained for both the gauge invariant large mass expansion as well as the short time Schwinger–DeWitt expansion. Final...

Quantum systems with external electromagnetic fields: The large mass asymptotics

The large mass asymptotics of the quantum evolution problem for a system of charged particles that mutually interact through scalar fields and couple to an arbitrary time‐varying external electromagnetic field is rigorously described. If K(x,t; y,s;m) denotes the coordinate space propagator (time evolution kernel) of this system, the singular perturbation behavior of K as mass m→∞ is expressed in terms of a gauge invariant asymptotic expansion. In terms of the external fields and interparticle interactions, this expansion provides a nonperturbative approximation for the propagator K that is valid for all particle coordinates x, y and for finite time displacements t−s. For the class of analytic scalar and vector fields that are defined as Fourier transforms of time‐dependent measures, the existence of this asymptotic series for K in powers of (m)−1 is established for both real and complex masses. Explicit bounds for the error term are obtained and a manifestly gauge invariant transport recurrence relation ...

Schrödinger spectral kernels: High order asymptotic expansions

The large energy behavior of the spectral kernel for the N‐body Schrodinger Hamiltonian is obtained. In a setting of a d‐dimensional Euclidean space without boundaries, the Schrodinger Hamiltonian H is the sum of the negative Laplacian plus a real‐valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N‐body problem, since they do not necessarily decrease as ‖x‖→∞. Let {e(x,y;λ): λ∈R} be the family of spectral kernels generated by H. In the λ→∞ limit, explicit higher order asymptotic expansions are obtained for e(x,y;λ) and its associated Riesz means. The asymptotic expansion is uniform in x and y and is accompanied by estimates of the error term.

Sum rules in classical scattering

This paper derives sum rules associated with the classical scattering of two particles. These sum rules are the analogs of Levinson’s theorem in quantum mechanics which provides a relationship between the number of bound‐state wavefunctions and the energy integral of the time delay of the scattering process. The associated classical relation is an identity involving classical time delay and an integral over the classical bound‐state density. We show that equalities between the Nth‐order energy moment of the classical time delay and the Nth‐order energy moment of the classical bound‐state density hold in both a local and a global form. Local sum rules involve the time delay defined on a finite but otherwise arbitrary coordinate space volume Σ and the bound‐state density associated with this same region. Global sum rules are those that obtain when Σ is the whole coordinate space. Both the local and global sum rules are derived for potentials of arbitrary shape and for scattering in any space dimension. Fina...

Integral bounds for N‐body total cross sections

We study the behavior of the total cross sections in the three‐ and N‐body scattering problem. Working within the framework of the time‐dependent two‐Hilbert space scattering theory, we give a simple derivation of integral bounds for the total cross section for all processes initiated by the collision of two clusters. By combining the optical theorem with a trace identity derived by Jauch, Sinha, and Misra, we find, roughly speaking, that if the local pairwise interaction falls off faster than r−3, then σtot(E) must decrease faster than E−1/2 at high energy. This conclusion is unchanged if one introduces a class of well‐behaved three‐body interactions.

Constructive representations of propagators for quantum systems with electromagnetic fields

The quantum evolution of an N‐body system of particles that mutually interact through scalar fields and couple to an arbitrary external electromagnetic field is rigorously described. Both operator and kernel valued solutions to the evolution problem are found. Based upon a particular realization of the Dyson expansion, a convergent series representation of the propagator (the kernel of the Schrodinger time evolution operator) is obtained. The basic approach is to embed the quantum evolution problem in the larger class of evolution problems that result if mass is allowed to be complex. Quantum evolution with real mass is considered to be the boundary value of the complex mass evolution problem. The constructive representation of the propagator is determined for the class of analytic scalar and vector fields that are given as Fourier transforms of time‐dependent scalar and vector‐valued measures.

An asymptotic analysis of quantum evolution with electromagnetic fields

A singular perturbation expansion of solutions to the Schrodinger initial value problem is constructed using an approximate propagator. For a nonrelativistic quantum system interacting with time‐dependent external electromagnetic fields, this approximate propagator defines a gauge invariant semiclassical expansion that is realized by large mass scaling. The asymptotic nature of this approximation is established by constructing error estimates that bound the Hilbert space norm difference between the exact and approximate evolved states. The maximum order of the approximation is determined explicitly as a function of the number of derivatives supported by the scalar and vector potentials. The asymptotic expansion is obtained when the configuration space Ω=Rd, and also for problems where Ω is a proper subset of Rd and the self‐adjoint Hamiltonian is defined using a supplementary boundary condition—typically Dirichlet or periodic.

Propagators for relativistic systems with non‐Abelian interactions

The relativistic evolution of a system of particles in the proper‐time Schwinger–DeWitt formalism is investigated. For a class of interactions that can be represented as Fourier transforms of bounded complex matrix‐valued measures, a Dyson series representation of the propagator is obtained. This class of interactions is non‐Abelian and includes both external electromagnetic and Yang–Mills fields. The study of the relativistic problem is facilitated by embedding the original quantum evolution into a larger class of evolution problems that result if one makes an analytic continuation of the metric tensor gμν. This continuation is chosen so that the extended propagator shares (for all signatures of gμν ) the Gaussian decay properties typical of heat kernels. Estimates of the nth‐order Dyson iterate kernels are found that ensure the absolute convergence of the perturbation series. In this fashion a number of analytic and smoothness properties of the propagator are determined. In particular, it is demonstrate...