Classical pseudo-differential calculus on $\mathbb{ R}^{ d}$ can be viewed as a (non-commutative) functional calculus for the standard position and momentum operators $(Q_{ 1}, \dots , Q_{ d})$ and $(P_{ 1}, \dots , P_{ d})$. We generalise this calculus to the setting of two $d$-tuples of operators $A=(A_{ 1}, \dots , A_{ d})$ and $B=(B_{ 1}, \dots , B_{ d})$ acting on a Banach space $X$ such that $iA_{ 1}, \dots , iA_{ d}$ and $iB_{ 1}, \dots , iB_{ d}$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{ isA_j}e^{ itA_k} = e^{ itA_k}e^{ isA_j}$, $e^{ isB_j}e^{ itB_k} = e^{ itB_k}e^{ isB_j}$, and $e^{ isA_j}e^{ itB_k} = e^{ -ist \delta_{ jk}} e^{ itB_k}e^{ isA_j}$ $(1\le j,k\le d)$. We show that the resulting calculus $a\mapsto a(A,B) \in \mathscr{ L}(X)$, initially defined for Schwartz functions $a\in \mathscr{ S}(\mathbb{ R}^{ 2d})$, extends to symbols in the standard symbol class $S^{ 0}$ of pseudo-differential calculus provided appropriate bounds can be established. We also prove a transference result that bounds the operators $a(A,B)$ in terms of the twisted convolution operators $C_{ \widehat{ a}}$ acting on $L^{ 2}(\mathbb{ R}^{ 2d};X)$. We apply these results to obtain $R$-sectoriality and boundedness of the $H^{ \infty}$-functional calculus (and even the H\"ormander calculus), for the abstract harmonic oscillator $L = \frac12\sum_{ j=1}^d (A_j^2+B_j^2)-\frac12d$.
Categories: math.FA math-ph math.MP