Brian Jefferies

The Weyl calculus and Clifford analysis

A note is made on the connection between Cliiord analysis and the Weyl functional calculus for an n-tuple of bounded selfadjoint operators which do not necessarily commute with each other.

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.

The Weyl calculus for hermitian matrices

The Weyl calculus is a means of constructing functions of a system of hermitian operators which do not necessarily commute with each other. This note gives a new proof of a formula, due to E. Nelson, for the Weyl calculus associated with a system of hermitian matrices.

Bilinear integration with positive vector measures

The integration of vector (and operator) valued functions with respect to vector (and operator) valued measures can be simplified by assuming that the measures involved take values in the positive elements of a Banach lattice.

Characterization of one-dimensional point interactions for the Schrödinger operator by means of boundary conditions

We find a new representation of all boundary conditions corresponding to the so-called point interactions. We show that there is one-to-one correspondence between one-dimensional point interactions and boundary conditions of the form with α,β,η satisfying |η| = |α|2 + |β|2 = 1.

Differential properties of the numerical range map of pairs of matrices

Abstract Let A = (A 1 , A 2 ) be a pair of Hermitian operators in C n and A = A1 + iA2. We investigate certain differential properties of the numerical range map n A : x → (〈A 1 x, x〉, 〈A 2 x, x〉) with the aim of better understanding the nature of the numerical range W(A) of A. For example, the joint eigenvalues of A correspond to the stationary points of n A (i.e. points where the derivative n′ A vanishes). Moreover, points x where rank n′ A (x) = 2 get mapped by n A into the interior W(A)° of W(A). For n = 2, it turns out that if A1 and A2 have no common invariant subspace, then the image under n A of the set Σ1(A) consisting of those points x with rank n′ A (x) = 1 is precisely the boundary ∂W(A) of W(A), and the image under n A of the rank 2 points for n′ A is precisely W(A)°; there are no rank 0 points for n′ A . As a consequence (for n = 2) we have that A1A2 = A2A1 iff Σ 1 (A) ≠ n A −1 (∂W(A)) .

Some Recent Applications of Bilinear Integration

Three topics featuring bilinear integration are described: the noncommutative Feynman-Kac formula, the connection between stationary state and time-dependent scattering theory and the stochastic integration of vectorvalued processes.

On the additivity of unbounded set functions

The set funcitions associated with Schrodingers equation are known to be unbouded on the algebra of cylinder sets. However, there do exist examples of scalar valued set functions which are unbounded, yet o-additive on the underlying algebra of sets. The purpose of this note is to show that the set functions associated with Schrodingers equation are not o-additive on cylinder sets. In the course of the proof, general conditions implying the non-o-additivity of underbounded set functions are given

Lacunas in the Support of the Weyl Calculus for Two Hermitian Matrices

The connection between Clifford analysis and the Weyl functional calculus for a d-tuple of bounded selfadjoint operators is used to prove a geometric condition due to J. Bazer and D. H. Y. Yen for a point to be in the support of the Weyl functional calculus for a pair of hermitian matrices. Examples are exhibited in which the support has gaps.

The propagator of the radial Dirac equation

The propagator for the radial Dirac equation is explicitly constructed. It turns out to be a distribution of order zero, but it is shown that there exists no path-space measure associated with this equation.