Observability inequality for the von Neumann equation in crystals (2512.10897v1)

Abstract: We provide a quantitative observability inequality for the von Neumann equation on $\mathbb{R}^d$ in the crystal setting, uniform in small $\hbar$. Following the method of Golse and Paul (2022) proving this result in the non-crystal setting, the method relies on a stability argument between the quantum (von Neumann) and classical (Liouville) dynamics and uses an optimal transport-like pseudo-distance between quantum and classical densities. Our contribution yields in the adaptation of all the required tools to the periodic setting, relying on the Bloch decomposition, notions of periodic Schrödinger coherent state, periodic Töplitz operator and periodic Husimi densities.

• The paper establishes a quantitative observability inequality for the von Neumann equation in crystals using Bloch decomposition and periodized optimal transport techniques.
• It leverages periodic coherent states, Töplitz quantization, and Husimi transforms to link quantum and classical dynamics uniformly in the semiclassical limit.
• The findings provide explicit operator-based bounds that have significant implications for quantum control and observability in periodic media.

Observability Inequality for the von Neumann Equation in Crystals

Introduction and Motivation

This work rigorously establishes a quantitative observability inequality for the von Neumann equation in the context of periodic quantum systems, specifically crystals, and demonstrates uniformity in the semiclassical limit ($\hbar \to 0$). The analysis extends the method of optimal transport pseudo-distances, originally developed for non-periodic systems by Golse and Paul [golse2022quantitative], to periodic settings via Bloch decomposition and periodic generalizations of coherent states, Töplitz quantization, and Husimi transforms.

The motivation lies in controllability and observability problems for quantum systems, where the von Neumann equation describes the unitary evolution of quantum density matrices, which is the physically relevant framework for infinite periodic systems such as crystals, where the wavefunction formalism is inadequate due to the nonlocality and infinite dimensionality of the electronic system.

Mathematical Framework

Periodic Setting and Bloch Decomposition

The analysis is set on a full-rank Bravais lattice $L \subset \mathbb{R}^d$ with associated unit cell $\Gamma$ and reciprocal lattice $L^*$ with unit cell $\Gamma^*$. The periodicity allows the introduction of the Bloch transform, which decomposes the Hilbert space as a direct integral over quasi-momenta $k \in \Gamma^*$. All operators of interest (Hamiltonians, observables, density matrices) commute with translations by $L$ and hence admit block-diagonal structure under the Bloch decomposition. The physical density is then recast as a density per unit cell using a periodic trace.

Quantum and Classical Dynamics

The quantum evolution is governed by the von Neumann equation for a periodic Hamiltonian $H = -\frac{\hbar^2}{2}\Delta + V$, $V$ being $L$-periodic and Lipschitz. The system evolves density matrices $R(t)$ that are non-negative, self-adjoint, and commute with lattice translations. The classical counterpart is the Liouville equation evolving densities $f(t, x, \xi)$ on phase space, periodic over $x$. Both dynamics are connected via semiclassical analysis.

Quantum-Classical Optimal Transport and Observability

Optimal Transport Pseudo-Distance in Periodic Setting

To relate classical and quantum observability, the paper introduces a periodized version of the quantum-classical optimal transport pseudo-metric, following [golse2022quantitative]. The main technical challenge is adapting the transportation cost and coupling constraints to periodic geometry. Notably, the cost function is periodized via projection to the unit cell and regularized for operator-theoretic compatibility.

Given a pair $(f, R)$, the cost involves the quadratic distance $|P_\Gamma(x-y)|^2$ between phase space points $x, y$, together with momentum terms, and is evaluated over admissible operator-valued couplings $Q(x,\xi)$ whose marginals recover $f$ and $R$ under integration and partial trace.

Periodic Töplitz and Husimi Transforms

The mapping from classical to quantum densities is via a periodic Töplitz quantization, and the reverse map, from quantum to classical, uses the periodic Husimi transform. Coherent states and their periodic generalizations play a key role in characterizing quantum localization in phase space and are central to establishing quantitative bounds.

Main Results

Observability Inequality in Crystals

The central result is a quantitative observability inequality for the von Neumann equation in the periodic crystal setting, formulated as follows. For a periodic density matrix $R(t)$ solving the von Neumann equation with periodic Hamiltonian, the inequality is:

$$\int_0^T \operatorname{Tr} (1_{\Omega^L_\delta} R(t) 1_{\Omega^L_\delta}) dt \geq C_{GC} \int_{K} \text{symbol} + \text{error terms}$$

where $\Omega \subset \Gamma$ is the observation region, $K\subset \Gamma \times \mathbb{R}^d$ is a compact set in phase space, and $C_{GC}$ is controlled by a geometric control condition on classical trajectories. The error terms are uniform in $\hbar$ under suitable conditions.

Strong Claims

• The observability estimate holds uniformly in the semiclassical parameter $\hbar$ as $\hbar\to 0$, up to explicit error terms.
• Both the periodic Töplitz and pure state (rank-1) initial data cases are addressed, with bounds depending on specific functionals (e.g., the standard deviation and $L^4$-norms of initial states).
• The periodic quantum observability constant only depends on geometric features (via the lattice and observation region) and analytic parameters (potential regularity, observation time).

Stability Between Quantum and Classical Dynamics

A crucial step is a stability estimate between quantum and classical flow in the optimal transport pseudo-metric, establishing an exponential growth bound (with rate determined by potential regularity and lattice geometry) of the distance between quantum and classical mass at the level of density operators and Liouville densities.

$$E_{\hbar, \lambda}(f(t), R(t)) \leq E_{\hbar, \lambda}(f^{\text{in}}, R^{\text{in}}) \exp(\eta_{\lambda, V} t)$$

where $E_{\hbar, \lambda}$ is the periodic quantum-classical pseudo-distance.

Periodic Coherent States and Operator Bounds

The paper develops a suite of operator-theoretic results for periodized coherent states, the properties of Töplitz quantized operators, and the periodic Husimi density. Specifically, upper bounds for the quantum-classical pseudo-distance between periodic Töplitz operators and their symbols, and between periodic pure states and their Husimi densities, are established. These are essential for quantifying initial discrepancies in the stability analysis.

Implications and Discussion

Practical and Theoretical Impact

• Controllability and Observability in Crystals: The result provides explicit, semiclassically uniform bounds for controlling or observing quantum states in periodic systems, which has direct implications for quantum control in solid-state physics and quantum optimal transport.
• Quantum-Classical Correspondence: The approach provides a quantitative bridge between quantum and classical observability, clarifying the relationship in the presence of periodic structures where semiclassical analysis is nontrivial.
• Operator Theory in Periodic Media: The construction and analysis of periodic coherent states, Töplitz, and Husimi transforms extend the operator-theoretic toolkit for periodic PDE analysis and quantum mechanics on lattices.

Outlook and Future Directions

• General Initial States: The results identify that Töplitz-type mixed states yield stronger quantitative bounds compared to periodic pure states under this framework, suggesting avenues for investigating more general classes of initial quantum data.
• Beyond Crystals: The techniques, relying on the Bloch transform and operator-valued optimal transport, may be adapted to other settings involving quasi-periodicity, disorder, or multi-band systems.
• Nonlinear Extensions: The methods could facilitate analysis in nonlinear quantum equations (e.g., Hartree-Fock or density functional theory in periodic settings), subject to adapting the Lie algebraic structure.

Conclusion

This paper provides a rigorous and quantitative analysis of observability for the von Neumann equation in crystal geometries, introducing a semiclassically uniform inequality that leverages optimal transport tools, the Bloch transform, and periodized operator calculus. The methodological contributions—periodized quantum-classical couplings, operator-valued cost functions, and sharp trace-norm observability estimates—open new directions for control and analysis of quantum systems in periodic media and reinforce the utility of optimal transport approaches in quantum PDEs (2512.10897).