Symbol Correspondence for Euclidean Systems

The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\star$-product of phase space functions. In this article, the $\star$-product of functions on the Euclidean motion group of rank three, $\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\star_s$ on $\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.

Boundary Value Problems at Infinite Genus

Boundary value problems are formulated on infinite-genus surfaces. These are solved for a variety of boundary conditions. The symbol calculus for differential operators is developed further for solution of parabolic differential equations at infinite genus.

TOEPLITZ OPERATORS AND THE ESSENTIAL BOUNDARY ON POLYANALYTIC FUNCTIONS

Let j be a nonzero integer and let U be a bounded domain. We construct a Fredholm symbol calculus for the C*-algebra generated by the poly-Bergman projection and the operators of multiplication by continuous functions. We define the j-removal boundary in Hilbert spaces of polyanalytic functions and prove that the quotient poly-Toeplitz C*-algebra generated by cosets of poly-Toeplitz operators with continuous symbols is *-isomorphic to the C*-algebra of continuous functions over the j-essential boundary. Unlike the Bergman case, we also show that if j ≠ ±1 then the j-essential boundary coincides with the set of non-isolated points on the boundary.