Two-Microlocal Wigner Measures Overview

• Two-microlocal Wigner measures are refined phase space tools that add a transverse variable to capture fine-scale concentration and dispersive effects in high-frequency regimes.
• They extend classical Wigner measures by distinguishing between absolutely continuous and singular components, thereby enhancing the analysis of propagation and observability in integrable systems.
• This framework provides precise propagation laws and operator-valued decompositions that underpin unique continuation and energy spreading in both torus and disk settings.

A two-microlocal Wigner measure is a refined phase space measure that captures both localization and concentration phenomena of high-frequency solutions to semiclassical PDEs near invariant submanifolds (typically Lagrangian tori) in completely integrable systems. This construction arises in the analysis of long-time semiclassical limits where standard semiclassical (Wigner) measures fail to distinguish between Lebesgue and singular components supported on resonant or rational invariant tori. Two-microlocalization introduces an additional "velocity" or "transverse" variable, enabling detection of fine-scale concentration and dispersive effects that dictate propagation, regularity, and observability properties in high-frequency quantum dynamics, especially for Schrödinger flows on tori and billiards on the disk (Anantharaman et al., 2014, Anantharaman et al., 2014, Anantharaman et al., 2015).

1. Definition and Construction of Two-Microlocal Wigner Measures

Let $H(\xi)$ be a completely integrable Hamiltonian on $(T^*\mathbb{T}^d,dx\,d\xi)$, and fix a resonant sublattice $\Lambda\subset\mathbb{Z}^d$ together with a point $\xi_0$ on the resonant manifold $$I_\Lambda = \{\xi\in(\mathbb{R}^d)^* : k\cdot\partial H(\xi)=0\ \forall k\in\Lambda\}$$. In a neighborhood of $\xi_0$, set $\xi = \sigma + \eta$ with $\sigma\in I_\Lambda$, $\eta\in\langle\Lambda\rangle$.

Given an $L^2$-normalized family $(u_h)$ oscillating at scale $h$, the two-microlocal Wigner distribution is built by zooming in transversally to $I_\Lambda$ at scale $1/\tau_h$, typically with $\tau_h\to\infty$ as $h\to0$ (critical case $\tau_h \sim h^{-1}$). Test symbols $$a(x,\xi,\eta)\in C^\infty_c(T^*\mathbb{T}^d\times\langle\Lambda\rangle)$$ are homogeneous of degree zero in $\eta$ for $|\eta|\gg1$ and localized in Fourier-$x$ to modes in $\Lambda$. The quantization

$$\mathrm{Op}_h^\Lambda(a) := \mathrm{Op}_h\big(a(x,\xi,\tau_h\eta(\xi))\big)$$

acts on $L^2(\mathbb{T}^d)$. The two-microlocal Wigner distribution at time $t$ is defined by

$$\langle w_h^\Lambda(t), a\rangle = \langle u_h, S_h(\tau_h t)^*\mathrm{Op}_h^\Lambda(a) S_h(\tau_h t)u_h\rangle_{L^2(\mathbb{T}^d)},$$

where $S_h(t) = \exp(-i t H(h D_x)/h)$.

Weak-* limits (in $h\to0$, $R\to\infty$) of these distributions yield two positive measures:

• $\tilde\mu^\Lambda(t,dx,d\xi,d\eta)$, supported on $\xi\in I_\Lambda$ and $|\eta|\to\infty$
• $\tilde\mu_\Lambda(t,dx,d\xi,d\eta)$, supported on $\xi\in I_\Lambda$ and finite $\eta$.

On the disk (billiard system), the construction is analogous: for a rational torus $I_{\alpha_0}$ defined via the angle of incidence $\alpha_0=\pi p/q$, one introduces adapted action–angle coordinates $(s,\theta,E,J)$, and the transverse variable $\eta=J'/h$ with $J'=J+E\sin\alpha_0$ (Anantharaman et al., 2014, Anantharaman et al., 2015).

2. Propagation Laws and Invariance

The two-microlocal measures satisfy specific propagation (transport) invariance laws reflecting the flows tangent and transverse to $I_\Lambda$:

• The measure $\tilde\mu^\Lambda$ is invariant under both
  • the classical flow $\varphi^0_s(x,\xi,\eta) = (x+s\,dH(\xi),\xi,\eta)$
  • the transverse linearized/second-order flow $$\varphi^1_s(x,\xi,\eta) = (x+s\,d^2 H(\sigma)\,\eta/|\eta|,\xi,\eta)$$.
• At the critical scale $\tau_h=h^{-1}$, the second-microlocal measure on the disk projects to an operator-valued measure on angular fibers, evolving under a 1D Heisenberg (Schrödinger) equation along the periodic direction, enforcing additional regularization (Anantharaman et al., 2014, Anantharaman et al., 2014).

3. Regularity Thresholds and Dichotomy

A fundamental feature of two-microlocal Wigner measures is the emergence of a critical time scale $T_h = h^{-1}$ separating regimes of possible singular or regular semiclassical measures in position. For timescales $\tau_h\ll h^{-1}$, arbitrary invariant (possibly singular) measures arise. For $\tau_h\gtrsim h^{-1}$, and $d^2 H(\xi)$ definite, all semiclassical measures project to absolutely continuous densities in configuration space $x$:

• If $\tau_h\ll h^{-1}$, measures singular in $x$ persist;
• If $\tau_h \simeq h^{-1}$ or $\tau_h\gg h^{-1}$, all measures are absolutely continuous in $x$, reflecting dispersive smoothing induced by the Schrödinger flow (Anantharaman et al., 2014).

On the disk, the analogous threshold manifests as absolute continuity in the angular variable on rational tori; semiclassical mass must spread in the periodic direction due to Heisenberg propagation, precluding localization on individual periodic orbits (Anantharaman et al., 2014, Anantharaman et al., 2015).

4. Structure Theorems and Delocalization

Both on the torus and the disk, the global structure of high-frequency limits is given by decomposing the full semiclassical measure as a sum over rational invariant tori and the Lebesgue component:

• For irrational flows, uniquely ergodic behavior enforces Lebesgue measure on tori.
• For rational tori, the second-microlocal measure is encoded by a finite positive trace-class operator-valued measure $\rho_\Lambda$ (resp. $\rho_{\alpha_0}$ on the disk), yielding absolutely continuous projections onto base variables.
• On the disk, each non-Lebesgue term $\nu_{\alpha_0}$ in the decomposition is absolutely continuous in $(s,\theta)$, with an explicit operator-valued description governed by commutation with a 1D Hamiltonian $P_{\alpha_0, \omega}$ on Floquet fibers (Anantharaman et al., 2015).

This decomposition enables sharp delocalization results: limit measures of $|u_h|^2\,dz$ are absolutely continuous in the interior (and even at the boundary, for appropriate boundary data), and no concentration of energy occurs on periodic orbits except the boundary itself.

5. Observability and Unique Continuation

The two-microlocal calculus facilitates the proof of observability inequalities and unique continuation properties for high-frequency solutions. On the torus, the following estimate holds for $u\in L^2(\mathbb{T}^d)$, $U\subset \mathbb{T}^d$ open, and $\chi$ localizing away from degenerate directions:

$$\| \chi(h D_x)u \|_{L^2}^2 \leq C \int_0^T \|1_U S_h^{t/h} \chi(h D_x)u \|_{L^2}^2 dt.$$

This directly links the microlocal decomposition to the impossibility of invariant measures vanishing on $U \times \operatorname{supp} \chi$ (Anantharaman et al., 2014).

On the disk, for any open set $\Omega\subset\mathbb{D}$ intersecting the boundary (or any open arc $\Gamma$ of the boundary), corresponding $L^2$-norm or $H^1$-norm inequalities preclude energy concentration on periodic orbits. The two-microlocal analysis reduces global observability to unique continuation for 1D Schrödinger operators on the circle, enforced by operator-valued invariance and absolute continuity of projected measures (Anantharaman et al., 2014, Anantharaman et al., 2015).

6. Generalization to Compact Integrable Systems

The two-microlocal construction extends to arbitrary compact integrable quantum systems where the principal symbols $A_1,\ldots,A_d$ of a commuting family of $h$-PDOs define a joint integrable system. Using action–angle variables, the local structure and propagation properties established for tori generalize via patching over the invariant tori, leading to the same regularity, propagation, and observability results as on the standard torus (Anantharaman et al., 2014).

7. Summary Table: Key Features of Two-Microlocal Wigner Measures

Feature	Torus/General System	Disk
Additional Variable	Transversal $\eta$ to $I_\Lambda$	Action–angle deviation $\eta$
Critical Regularity Threshold	$\tau_h = h^{-1}$	$h^2$-quasimode accuracy
Structure of Rational Torus Component	Trace-class operator-valued bundle	Trace-class operator-valued measure on $L^2(\theta)$
Propagation Mechanism	Classical + second-order flows	1D Heisenberg evolution on circle
Observability Implication	No vanishing invariant measure on open sets	No energy concentration on periodic orbits except boundary

The two-microlocal Wigner measure formalism provides a unified analytical tool for diagnosing fine-scale concentration and regularity phenomena in the high-frequency limit of quantum completely integrable systems, with far-reaching consequences for understanding wave propagation, dispersion, regularity, and the limits of controllability and observation in both quantum and classical dynamics (Anantharaman et al., 2014, Anantharaman et al., 2014, Anantharaman et al., 2015).