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The purpose of this note is to review some recent results concerning the pseudospectra and the eigenvalues asymptotics of non-selfadjoint semiclassical pseudo-differential operators subject to small random perturbations., Comment: A short review article

It has been known since the work of Avakumov\'ic, H\"ormander and Levitan that, on any compact smooth Riemannian manifold, if $-\Delta_g \psi_\lambda = \lambda \psi_\lambda$, then $\|\psi_\lambda\|_{L^\infty} \leq C \lambda^{\frac{d-1}{4}} \|\psi_\lambda\|_{L^2}$. It is believed that, on manifolds of negative curvature, such a bound can be largely improved; however, only logarithmic improvements in $\lambda$ have been obtained so far. In the present paper, we obtain polynomial improvements over the previous bound in a generic setting, by adding a small random pseudodifferential perturbation to the Laplace-Beltrami operator., Comment: 24 pages, minor corrections

We consider small random perturbations of the standard high-symmetry tunneling potentials in the Bistritzer-MacDonald Hamiltonian describing twisted bilayer graphene. Using methods developed by Sj\"ostrand for studying the spectral asymptotics of non-selfadjoint pseudo-differential operators, we prove that for sufficiently small twisting angles the Hamiltonian will not exhibit a flat band with overwhelming probability, and hence the absence of the so-called \textit{magic angels}. Moreover, we prove a probabilistic Weyl law for the eigenvalues of the non-selfadjoint tunneling operator, subject to small random perturbations, of the Bistritzer-MacDonald Hamiltonian in the chiral limit., Comment: 32 pages, 1 figure

Why do experiments only exhibit one magic angle if the chiral limit of the Bistritzer-MacDonald Hamiltonian suggest a plethora of them? - In this article, we investigate the remarkable stability of the first magic angle in contrast to higher (smaller) magic angles. More precisely, we examine the influence of disorder on magic angles and the Bistritzer-MacDonald Hamiltonian. We establish the existence of a mobility edge near the energy of the flat band for small disorder. We also show that the mobility edges persist even when all global Chern numbers become zero, leveraging the $C_{2z}T$ symmetry of the system to demonstrate non-trivial sublattice transport. This effect is robust even beyond the chiral limit and in the vicinity of perfect magic angles, as is expected from experiments.

We study the long time Schr\"odinger evolution of Lagrangian states $f_h$ on a compact Riemannian manifold $(X,g)$ of negative sectional curvature. We consider two models of semiclassical random Schr\"odinger operators $P_h^\alpha=-h^2\Delta_g +h^\alpha Q_\omega$, $0<\alpha\leq 1$, where the semiclassical Laplace-Beltrami operator $-h^2\Delta_g$ on $X$ is subject to a small random perturbation $h^\alpha Q_\omega$ given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of $Q_\omega$ is bounded, but oscillates and decorrelates at scale $h^{\beta}$, $0< \beta < \frac{1}{2}$. We prove a quantitative result that, under appropriate conditions on $\alpha,\beta$, in probability with respect to $\omega$ the long time propagation $$\mathrm{e}^{\frac{i}{h}t_h P_h^\alpha } f_h, \quad o(|\log h|)=t_h\to\infty, ~~h\to 0,$$ rescaled to the local scale of $h$ around a uniformly at random chosen point $x_0$ on $X$, converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and $\omega$-almost sure version of this convergence along sufficiently fast decaying subsequences $h_j\to 0$., Comment: 69 pages, 3 figures, version 2

We study the small singular values of the $2$-dimensional semiclassical differential operator $P = 2\,\mathrm{e}^{-\phi/h}\circ hD_{\overline{z}}\circ \mathrm{e}^{\phi/h}$ on $S^1+iS^1$ and on $S^1+i\mathbb{R}$ where $\phi$ is given by $\sin y$ and by $y^3/3$, respectively. The key feature of this model is the fact that we can pinpoint precisely where in phase space the Poisson bracket $\{p,\overline{p}\}=0$, where $p$ is the semiclassical symbol of $P$. We give a precise asymptotic description of the exponentially small singular values of $P$ by studying the tunneling effects of an associated Witten complex. We use these asymptotics to determine a Weyl law for the exponentially small singular values of $P$., Comment: 24 pages

We prove localization with high probability on sets of size of order $N/\log N$ for the eigenvectors of non-Hermitian finitely banded $N\times N$ Toeplitz matrices $P_N$ subject to small random perturbations, in a very general setting. As perturbation we consider $N\times N$ random matrices with independent entries of zero mean, finite moments, and which satisfy an appropriate anti-concentration bound. We show via a Grushin problem that an eigenvector for a given eigenvalue $z$ is well approximated by a random linear combination of the singular vectors of $P_N-z$ corresponding to its small singular values. We prove precise probabilistic bounds on the local distribution of the eigenvalues of the perturbed matrix and provide a detailed analysis of the singular vectors to conclude the localization result., Comment: Minor corrections and reorganization

We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex $N\times N$ matrices subject to small random perturbations. We show that with probability close to $1$ this log-potential is, up to a small error, determined by the singular values of the unperturbed matrix which are larger than some small $N$-dependent cut-off parameter.

We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean $0$, variance $1$ and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by T. Christiansen and M. Zworski., Comment: 47 pages, 1 figure

We study the spectra of general $N\times N$ Toeplitz matrices given by symbols in the Wiener Algebra perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove an asymptotic formula for the number of eigenvalues of the perturbed matrix in smooth domains. We show that these eigenvalues follow a Weyl law with probability sub-exponentially close to $1$, as $N\gg1$, in particular that most eigenvalues of the perturbed Toeplitz matrix are close to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix., Comment: 23 pages, 2 figures

We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix., Comment: 40 pages, 4 figures

We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} \| \chi (-h^2\Delta + V-\lambda^2)^{-1} \chi \|_{L^2\to H^1} \leq C \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $\lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2\Delta + V-\lambda^2)^{-1}$ as an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} \mathrm{Im} \lambda \leq - C^{-1} \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-\Delta u = Vu$ with $0\not\equiv V\in L^{\infty}(\mathbb{R}^d)$. We show that there exist a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has \begin{equation*} \int_{B(0,R+1)\backslash \overline{B(0,R)}}|u(x)|^2 dx \geq M^{-1}R^{-4/3} \mathrm{e}^{-M \|V\|_{\infty}^{2/3} R^{4/3}}\|u\|^2_2. \end{equation*}, Comment: 21 pages

We consider a class of one-dimensional nonselfadjoint semiclassical pseudo-differential operators, subject to small random perturbations, and study the statistical properties of their (discrete) spectra, in the semiclassical limit $h\to 0$. We compare two types of random perturbation: a random potential vs. a random matrix. Hager and Sj\"ostrand had shown that, with high probability, the local spectral density of the perturbed operator follows a semiclassical form of Weyl's law, depending on the value distribution of the principal symbol of our pseudodifferential operator. Beyond the spectral density, we investigate the full local statistics of the perturbed spectrum, and show that it satisfies a form of universality: the statistical only depends on the local spectral density, and of the type of random perturbation, but it is independent of the precise law of the perturbation. This local statistics can be described in terms of the Gaussian Analytic Function, a classical ensemble of random entire functions., Comment: 81 pages, 7 figures

We study the spectrum of large a bi-diagonal Toeplitz matrix subject to a Gaussian random perturbation with a small coupling constant. We obtain a precise asymptotic description of the average density of eigenvalues in the interior of the convex hull of the range symbol., Comment: 26 pages, 1 figure

This is a first paper by the authors dedicated to the distribution of eigenvalues for random perturbations of large bidiagonal Toeplitz matrices., Comment: 34 pages, 4 figures

We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E.B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to $1$, most of the eigenvalues are close to a circle. We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description., Comment: 4 figures

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to random perturbations with a small coupling constant $\delta$. Assume that $\exp(-\frac{1}{Ch}) < \delta \ll h^{\kappa}$ for constants $C,\kappa>0$ suitably large. Let $\Sigma$ be the closure of the range of the principal symbol. We study the $2$-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator $P_h^{\delta}$ and prove an $h$-asymptotic formula for the average $2$-point density of eigenvalues. With this we show that two eigenvalues of $P_h^{\delta}$ in the interior of $\Sigma$ exhibit close range repulsion and long range decoupling., Comment: 46 pages, 4 figures, Commun. Math. Phys. (2016)

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h\rightarrow 0$) subject to small random perturbations. Furthermore, we let the coupling constant $\delta$ be $\exp\{-\frac{1}{Ch}\}\leq \delta \ll h^{\kappa}$ for constants $C,\kappa>0$ suitably large. Let $\Sigma$ be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sj\"ostrand show that if $\delta \gg\exp\{-\frac{1}{Ch}\}$ there is, with a probability close to $1$, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance $\gg\left(-h\ln{\delta h}\right)^{\frac{2}{3}}$ to the boundary of $\Sigma$. We study the intensity measure of the random point process of eigenvalues and prove an $h$-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in $\Sigma$: The interior of the of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly., Comment: 75 pages, 13 figures