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We consider operators acting on a UMD Banach lattice $X$ that have the samealgebraic structure as the position and momentum operators associated with theharmonic oscillator $-\frac12\Delta + \frac12|x|^{2} $ acting on$L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonicoscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ fortuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where$iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy thecanonical commutator relations. We prove functional calculus results for theseabstract harmonic oscillators that match classical H\"ormander spectralmultiplier estimates for the harmonic oscillator $-\frac12\Delta +\frac12|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where theunderlying metric measure space is not doubling and the use of function spacesthat are not particularly well suited to extrapolation arguments. For instance,as an application we treat the harmonic oscillator on mixed norm Bargmann-Fockspaces. Our approach is based on a transference principle for the Schr\"odingerrepresentation of the Heisenberg group that allows us to reduce the problem tothe study of the twisted Laplacian on the Bochner spaces$L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of theStone-von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces.

Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices