Hörmander’s famous Fourier multiplier theorem ensures the $L_p$-boundedness of $F(-{\Delta }_{ℝ}D)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s\>\frac{D}{2}$, where we denote by $\mathscr{H}\left(s\right)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A\ge 0$ and yield the $L_p$-boundedness of $F\left(A\right)$ provided $F\in \mathscr{H}\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta }_{ℝ}+x^2$ shows that in general $s\>\frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In this paper,...