[Paper Review] Eigenvalue interaction for a class of non-selfadjoint operators under random perturbations

Martin Vogel|arXiv (Cornell University)|Dec 1, 2014

Spectral Theory in Mathematical Physics4 references3 citations

TL;DR

This paper studies the eigenvalue distribution of a non-selfadjoint $h$-differential operator $P_h$ under small random perturbations with coupling constant $\delta$. Using semiclassical analysis and random point process theory, it derives an $h$-asymptotic formula for the average 2-point density, demonstrating eigenvalue repulsion at short ranges and decoupling at long ranges within the numerical range $\Sigma$. This reveals universal statistical behavior in the spectral response to noise.

ABSTRACT

We consider a non-selfadjoint $h$-differential model operator $P_h$ in the semiclassical limit ($h ightarrow 0$) subject to random perturbations with a small coupling constant $\delta$. Assume that $\exp(-\frac{1}{Ch}) 0$ 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.

Motivation & Objective

• To understand the statistical behavior of eigenvalues of non-selfadjoint operators under small random perturbations.
• To derive an asymptotic formula for the average 2-point density of eigenvalues in the semiclassical limit ($h \to 0$).
• To characterize the interaction between eigenvalues in the interior of the numerical range $\Sigma$ under random perturbations.
• To establish universal spectral statistics—repulsion at short range and decoupling at long range—via random matrix-like behavior.

Proposed method

• Modeling the operator $P_h$ as a semiclassical $h$-differential operator with non-selfadjoint structure.
• Introducing a small random perturbation with coupling constant $\delta \ll 1$ to study spectral fluctuations.
• Analyzing the $2$-point intensity measure of the random eigenvalue point process associated with $P_h^\delta$.
• Applying semiclassical analysis and asymptotic methods to derive an $h$-dependent formula for the average 2-point density.
• Using the principal symbol's range $\Sigma$ as the domain for eigenvalue localization and interaction analysis.
• Establishing asymptotic control on eigenvalue correlations through $h$-dependent estimates in the limit $h \to 0$.

Experimental results

Research questions

• RQ1 How do eigenvalues of a non-selfadjoint $h$-differential operator behave under small random perturbations?
• RQ2 What is the asymptotic form of the average 2-point density of eigenvalues as $h \to 0$?
• RQ3 Do eigenvalues exhibit repulsion at short distances and decoupling at long distances in the interior of the numerical range $\Sigma$?
• RQ4 Can the spectral statistics of $P_h^\delta$ be described by a universal $h$-asymptotic formula?

Key findings

• An $h$-asymptotic formula is derived for the average 2-point density of eigenvalues of $P_h^\delta$ in the semiclassical limit.
• Eigenvalues in the interior of $\Sigma$ exhibit strong short-range repulsion, indicating level repulsion similar to random matrix theory.
• At long ranges, eigenvalue pairs decouple, meaning their correlation vanishes asymptotically as $h \to 0$.
• The interaction structure is universal: repulsion dominates near eigenvalues, while distant pairs become statistically independent.
• The results hold under the condition $\delta \gg \exp(-1/(Ch))$ for sufficiently large $C > 0$, ensuring perturbation strength is above the tunneling threshold.