The Fefferman-Phong uncertainty principle for representations of Lie groups and applications

We prove a new uncertainty principle for square-integrable irreducible unitary representations of connected Lie groups. The concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local Muckenhoupt class $A_{\infty,{\rm loc}}$ associated with a subRiemannian left-invariant metric and a relatively invariant measure. The result is reminiscent of the Fefferman-Phong uncertainty principle, and is new even for the Schr\"odinger representation of the reduced Heisenberg group, which corresponds to the short-time Fourier transform. As an application, we give an optimal estimate of the order of magnitude of the bottom of the spectrum and of the essential spectrum of semiclassical anti-Wick operators in $\mathbb{R}^d$ with a nonnegative symbol $a$ in the class $A_{\infty}$ (in particular, for polynomial symbols). Precisely, we show that the infimum $\inf _{(x_0,\omega_0)\in{\mathbb{R}^{2d}}} -\!\!\!\!\!\int_{B((x_0,\omega_0),\sqrt{h})} a(x,\omega)\, dx\,d\omega$ represents (up to multiplicative constants) both a lower bound and an upper bound for the bottom of the spectrum, uniformly with respect to $h>0$. Similarly the quantity $$\liminf_{(x_0,\omega_0)\to\infty} -\!\!\!\!\!\!\int_{B((x_0,\omega_0),\sqrt{h})} a(x,\omega)\, dx\,d\omega$$ represents both a lower bound and an upper bound for the bottom of the essential spectrum, uniformly with respect to $h>0$. Similar results are proved for semiclassical symbol classes.