$\mathcal{R}$-boundedness of Riesz transforms and spectral multipliers of Schrödinger operators

Let $\text{LL}=-\mathrm{\Delta }+V$ be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform $\mathrm{\nabla }{\text{LL}}^{-\text{sfrac}12}$ is bounded on $L^p({\text{RR}}^d)$ for $p\in (1,2]$, with a bound independent of $V$. We prove more, namely that the set $$\{\mathrm{\nabla }{\text{LL}}^{-\text{sfrac}12},0\le V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\text{RR}}^d)\}$$ is $\mathcal{R}$-bounded on $L^p({\text{RR}}^d)$ for $p\in (1,2]$. We also prove the $\mathcal{R}$-boundedness on $L^p({\text{RR}}^d)$, for $p\in (1,\mathrm{\infty })$, of the spectral multipliers $$m(\text{LL}),0\le V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\text{RR}}^d)$$ for any $m:[0,\mathrm{\infty })\to \text{CC}$ satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators $-\mathbf{\Delta }+\mathbf{V}$ on vector-valued spaces $L^p({\text{RR}}^d,{\mathrm{\ell }}^2)$. In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.