Can one use pseudo-differential calculus in situations where no Fourier transform is available? On a manifold, one can use the euclidean pseudo-differential calculus locally, but can one find a global analogue? In the simpler case of (radial) Fourier multiplier calculus, the answer is well known: one can think of this calculus as a functional calculus of the Laplacian, and then generalise it, by operator theoretic methods, to large classes of Laplace like operators acting on various Banach spaces. The quintessential example of this approach is given by Alan McIntosh’s construction of a holomorphic calculus for sectorial operators.
Generalising the full pseudo-differential calculus is more challenging, but an operator theoretic perspective is nonetheless available: the Weyl calculus of the (euclidean) position and momentum operators. In this talk, we see how this calculus can be extended to pairs of group generators acting on a Banach space, and satisfying the same algebraic commutator condition as the position and momentum operators. This provides a foundation for the study of stochastic or geometric analogues of the standard quantum harmonic oscillator .
This is joint work with Jan van Neerven (Delft).