Quantum–classical convergence of energy–velocity measures

Let ${\cal P}:=T^*{\bf T}$ be the classical phase space, with Hamiltonian $H(p,q):=\frac{1}{2}p^2+V(q)$ and Liouville measure $\lambda$. Let $\bar v:{\cal P}\to{\bf R}^d$ be the asymptotic velocity and define the classical energy–velocity map

$$A:=(H,\bar v):{\cal P}\to{\bf R}^{d+1}.$$

Let $\nu:=\lambda A^{-1}$. For the quantum band energies $E_n^\hbar(k)$ and quantum asymptotic velocities $\bar v_n^\hbar(k)$, define

$$A^\hbar(n,k):=(E_n^\hbar(k),\bar v_n^\hbar(k)),$$

where ${\cal P}^\hbar:={\bf N}\times{\bf T}^*$ carries the semiclassical measure $\lambda^\hbar:=(2\pi\hbar)^d\mu_1\times\mu_2$, and let $\nu^\hbar:=\lambda^\hbar(A^\hbar)^{-1}$. Quantum–classical measure convergence conjecture. For all ${\cal L}$-periodic potentials $V\in C^\infty({\bf R}^d,{\bf R})$,

$$w^*\!-\!\lim_{\hbar\searrow0}\nu^\hbar=\nu.$$

Equivalently, for every $f\in C^0_0({\bf R}^{d+1},{\bf R})$,

$$\lim_{\hbar\searrow0}\int_{{\bf R}^{d+1}}f(x)\,d\nu^\hbar(x)=\int_{{\bf R}^{d+1}}f(x)\,d\nu(x).$$

The source states that this was proved for smooth potentials producing integrable or ergodic motion, with related results cited for Coulombic periodic potentials; the assertion for all smooth periodic potentials is therefore the remaining general case.