Cwikel estimates and Weyl's laws for $L\log L$ potentials on quantum tori

Abstract

We prove an \(L\log L\)-endpoint Cwikel estimate on quantum tori $\mathbb T_θ^d$, i.e., for every \(d\geq 1\) and \(x\in L\log L(\mathbb T_θ^d)\), \begin{equation*} (1-Δ)^{-\frac{d}{4}}λ(x)(1-Δ)^{-\frac{d}{4}}\in\mathcal{L}_{1,\infty} \end{equation*} with a bound depending only on the dimension, where $λ(x)$ is the left multiplication by $x$. The main ingredient is an abstract weak Hilbert--Schmidt estimate for multiplication operators in finite von Neumann algebras whose \(L_2\)-spaces admit orthonormal bases of unitaries. When $θ=0$, our argument also gives a simpler proof of the $L\log L$ Cwikel estimate on classical tori. For $d\geq2$, we use the $C^*$-algebraic principal symbol map to prove Weyl's laws for finite sums \[ \sum_{\ell=1}^N S_\ell(1-Δ)^{-d/4}λ(x_\ell) (1-Δ)^{-d/4}T_\ell, \] where $x_\ell\in L\log L(\mathbb T_θ^d)$ and $S_\ell,T_\ell\inΠ$, the $C^*$-algebra generated by $λ(C(\mathbb T_θ^d))$ and Fourier multipliers with continuous symbols homogeneous of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with $L\log L$ potentials.