[Paper Review] Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

Mildred Hager, Johannes Sjoestrand|ArXiv.org|Jan 16, 2006

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper establishes that for general $h$-pseudodifferential operators on ${\bf R}^n$, small random perturbations—specifically, random integral operators—induce eigenvalue distributions that follow a Weyl law in the semi-classical limit $h \to 0$, with probability tending to 1. The result extends prior one-dimensional findings by relaxing assumptions on symbol analyticity and differential independence, using Hilbert-Schmidt random operators and Grushin problem techniques to derive spectral asymptotics.

ABSTRACT

We consider quite general $h$-pseudodifferential operators on $R^n$ with small random perturbations and show that in the limit of small $h$ the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.

Motivation & Objective

• To extend previous one-dimensional results on random perturbations of non-selfadjoint $h$-pseudodifferential operators to the multidimensional case.
• To weaken assumptions on the symbol, such as analyticity and independence of differentials, compared to earlier work.
• To show that eigenvalue distributions follow a Weyl law under small random perturbations, even when the unperturbed operator does not satisfy Weyl asymptotics.
• To establish that the probability of eigenvalue distribution following Weyl's law tends to 1 as $h \to 0$.
• To develop a framework using Grushin problems and random Hilbert-Schmidt operators to analyze spectral behavior under perturbation.

Proposed method

• Use of $h$-pseudodifferential operators with symbol space $S({\bf R}^{2n}, m)$ and order function $m$ satisfying polynomial growth conditions.
• Application of the Weyl quantization to define the operator $P$ on $L^2({\bf R}^n)$.
• Introduction of random perturbations via a random integral operator with i.i.d. Gaussian coefficients acting on an orthonormal basis.
• Employment of the Grushin problem framework to reduce spectral analysis to trace-class and Hilbert-Schmidt norm estimates.
• Use of determinant estimates for Gaussian random matrices and functional calculus to control eigenvalue distribution.
• Leverage of Hilbert-Schmidt random operator theory, including the invariance of distribution under orthonormal basis changes, to ensure almost sure convergence of spectral measures.

Experimental results

Research questions

• RQ1 Does the eigenvalue distribution of a non-selfadjoint $h$-pseudodifferential operator converge to a Weyl law under small random perturbations in higher dimensions?
• RQ2 Can the Weyl asymptotics be recovered in the semi-classical limit $h \to 0$ when the unperturbed operator does not satisfy Weyl's law?
• RQ3 How does the probability of eigenvalue distribution following Weyl's law behave as $h \to 0$ under random perturbations?
• RQ4 What is the role of the Hilbert-Schmidt structure of the random perturbation in ensuring spectral stability?
• RQ5 Can the spectral asymptotics be derived without assuming analyticity or independence of symbol differentials?

Key findings

• In the limit $h \to 0$, the eigenvalues of the perturbed operator are distributed according to the Weyl law with probability tending to 1.
• The result holds for a broad class of $h$-pseudodifferential operators on ${\bf R}^n$, without requiring analyticity of the symbol or independence of $dp$ and $d\overline{p}$.
• The random perturbation is given by a Hilbert-Schmidt operator with i.i.d. Gaussian coefficients, ensuring almost sure convergence of spectral measures.
• The Grushin problem technique reduces the spectral analysis to estimating determinants and Hilbert-Schmidt norms of random operators.
• The joint distribution of the random coefficients is invariant under orthonormal basis changes, enabling robust probabilistic estimates.
• The method establishes that the number of eigenvalues in a domain $\Omega \Subset {\bf C} \setminus \Sigma_\infty$ asymptotically matches the Weyl volume $\mathrm{vol}(p^{-1}(I))$ with high probability as $h \to 0$.