Lecture notes on Quantum Diffusion and Random Matrix Theory

Abstract: In joint work with Adam Black and Reuben Drogin, we develop a new approach to understanding the diffusive limit of the random Schrodinger equation based on ideas taken from random matrix theory. These lecture notes present the main ideas from this work in a self-contained and simplified presentation. The lectures were given at the summer school "PDE and Probability" at Sorbonne Universit\'e from June 16-20, 2025.

• The paper rigorously proves quantum diffusion in the random Schrödinger equation by synthesizing perturbative, harmonic analytic, and random matrix methods.
• It establishes precise operator norm bounds and spectral projection estimates that compare the effects of a random potential with free evolution.
• The analysis provides insights into delocalization, demonstrating diffusive scaling and effective random walk behavior in disordered media.

Quantum Diffusion and Random Matrix Theory: Rigorous Analysis of the Random Schrödinger Equation

Introduction and Motivation

The lecture notes present a rigorous framework for analyzing quantum diffusion in the random Schrödinger equation (RSE) using techniques from random matrix theory. The RSE, given by $i\partial_t \psi = \Delta \psi + \lambda V \psi$, models wave transport in disordered media, with $\Delta$ as the Laplacian, $V$ a random potential, and $\lambda \ll 1$ the coupling parameter. The central question is the characterization of the mean square displacement $r^2(t)$ of an initially localized wavefunction, which is directly related to the conductivity and localization properties of the underlying physical system.

The notes address the long-standing conjectures regarding the metal-insulator transition and the existence of extended states in higher dimensions ($d \geq 3$), as well as the scaling of localization length in $d=2$. The main technical achievement is a proof of quantum diffusion up to the diffusive time scale, leveraging an overview of perturbative, harmonic analytic, and random matrix methods.

Kinetic Time Scale Analysis

At times $t \ll \lambda^{-2}$, the evolution $e^{-itH}$ closely approximates the free evolution $e^{-it\Delta}$, with the random potential $V$ contributing only weak perturbations. The Duhamel formula is iterated to express $e^{-itH} - e^{-it\Delta}$ as a Dyson series, with each term $T_j(t)$ representing $j$-fold interactions with the potential.

A key technical tool is the non-commutative Khintchine inequality, which provides operator norm bounds for random matrices linear in the Gaussian randomness. For the first collision operator $T_1(t)$, this yields

$$\|T_1(t)\|_{\text{op}} \lesssim (\log \lambda^{-1}) \sqrt{t}$$

for $d \geq 3$, and an additional $\sqrt{\log t}$ factor for $d=2$. Higher-order terms $T_j(t)$ are controlled recursively, leading to

$$\|T_k(t)\|_{\text{op}} \lesssim (C k^{-1} (\log \lambda^{-1})^2 (t \log t))^{k/2}$$

with high probability.

These bounds enable precise control over spectral projections and resolvent operators. For smooth spectral windows of width $\delta$, one obtains

$$\|\chi_{\delta,E}(H) - \chi_{\delta,E}(\Delta)\|_{\ell^2 \to \ell^2} \lesssim \lambda \delta^{-1/2}$$

which is crucial for comparing the spectral statistics of $H$ and $\Delta$ down to scales $\delta \sim \lambda^2$.

Resolvent Bounds and Spectral Projections

The analysis extends to resolvent operators $R(z) = (H-z)^{-1}$, with $z = E + i\eta$. For $\eta \gtrsim \lambda^2$, the resolvent of $H$ is well-approximated by that of $\Delta$, and one can transfer $\ell^p \to \ell^q$ bounds via interpolation and duality. The Tomas-Stein restriction estimate is used to control spectral projections in Fourier space, yielding

$$\|R(E+i\eta)\|_{p \to q} \lesssim \lambda^{-c} (\lambda^2 \eta^{-1} + 1)$$

for $1 \leq p \leq 6/5$, $6 \leq q \leq \infty$, and $E$ away from critical values of the dispersion relation.

These bounds imply that eigenfunctions of $H$ are delocalized in $\ell^p$ norms for $p > 2$, with $\|\psi_k\|_{\ell^6} \lesssim \lambda$ for $d=2$.

Diffusive Time Scale and Quantum Diffusion

For times $t \gg \lambda^{-2}$, the potential $V$ induces strong scattering, and the evolution is governed by a kinetic equation for the phase space density. The effective dynamics correspond to a random walk with $\sim \lambda^2 t$ steps of size $\lambda^{-2}$, leading to the heuristic

$$|\psi_t(x)|^2 \sim (\lambda^{-2} t)^{-d/2} \exp(-\lambda^2 |x|^2 / t)$$

and $r(t) \sim \lambda^{-1} t^{1/2}$.

The analysis shifts to the resolvent $R(z)$, whose entries encode time-averaged information about the propagation of the wavefunction. The main technical result is a local law for the diagonal entries of the resolvent: $$|R_{00}(z) - \theta(z)| \leq \lambda^{1/2-\delta} (\lambda \eta^{-1})^3$$ for $\eta \gg \lambda^{2+1/6-\delta}$, where $\theta(z)$ solves the self-consistent equation

$$\theta = (\Delta - (z + \lambda^2 \theta))^{-1}_{00}$$

This is established via Gaussian concentration inequalities and a bootstrap argument, using the a priori $\ell^p \to \ell^q$ bounds.

Second Moment Analysis and Random Walk Representation

To capture the spatial spread of the resolvent, a self-consistent equation for the second moment $g(x) = (R F R^*)_{xx}$ is derived: $$g = \tilde{K} * f + \lambda^2 \tilde{K} * g + \mathfrak{e}_T$$ where $\tilde{K}(x) = |\tilde{M}_{0x}|^2$ and $\tilde{M} = (\Delta - (z + \lambda^2 \theta))^{-1}$. The solution is expressed as

$$g = (I - \lambda^2 K)^{-1} K f + (I - \lambda^2 K)^{-1} \mathfrak{e}_T$$

with $K$ the convolution operator by $\tilde{K}$. The Neumann series expansion of $(I - \lambda^2 K)^{-1}$ corresponds to the Green's function of a random walk with step distribution $\tilde{K}$.

Moment estimates for $\tilde{K}$ show that the walk has variance $\sim \lambda^{-4}$ per step, and the total spread after $n$ steps is $\sim n \lambda^{-4}$. Anticoncentration estimates for the random walk yield that the mass of $|R_{0x}|^2$ is concentrated on the diffusive scale $|x| \sim \lambda^{-1} \eta^{-1/2}$, with negligible mass near the origin.

Implications and Future Directions

The rigorous derivation of quantum diffusion up to the diffusive time scale in the random Schrödinger equation provides a solid foundation for understanding wave transport in disordered media. The synthesis of random matrix theory, perturbative expansions, and harmonic analysis yields robust operator norm and concentration bounds, enabling precise control over both spectral and spatial properties of the evolution.

The framework is sufficiently general to accommodate extensions to other random matrix ensembles and more complex models of disorder. The techniques developed here, particularly the use of self-consistent equations and non-commutative Khintchine inequalities, are likely to be applicable to the study of localization-delocalization transitions, mobility edges, and the fine structure of eigenvalue statistics in higher dimensions.

Open problems include the extension of these results to infinite volume, the rigorous characterization of the metal-insulator transition, and the analysis of higher-order moments and correlations. The interplay between random matrix theory and dispersive PDEs remains a fertile ground for further research.

Conclusion

These lecture notes provide a comprehensive and technically rigorous account of quantum diffusion in the random Schrödinger equation, leveraging advanced tools from random matrix theory to overcome longstanding obstacles in the field. The results establish delocalization and diffusive scaling up to the optimal time scale, with high-probability bounds and explicit operator norm estimates. The approach sets a new standard for the analysis of random operators in mathematical physics and opens avenues for future developments in the theory of disordered systems.