[Paper Review] On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus

Patrick Gérard, Thomas Kappeler+1 more|arXiv (Cornell University)|Jun 21, 2020

Advanced Mathematical Physics Problems8 references19 citations

TL;DR

This paper investigates the spectral properties of the Lax operator $L_u = D - T_u$ for complex-valued potentials $u \in H^{-s}_{c,0}(\mathbb{T})$ with $0 \leq s < 1/2$ on the torus. It proves that the eigenvalues $\lambda_n(u)$ are analytic in $u$, the gap lengths $\gamma_n(u) = \lambda_n(u) - \lambda_{n-1}(u) - 1$ lie in $\ell^{1,1-2s}(\mathbb{N}, \mathbb{C})$, and the map $\Gamma: u \mapsto (\gamma_n(u))_{n \geq 1}$ is analytic, establishing the moment map as a holomorphic object in this setting.

ABSTRACT

We investigate the spectrum of the Lax operator $L_u$ of the Benjamin-Ono equation on the torus for complex valued potentials $u$ in the Sobolev space $H^{-s}(\\mathbb{T},\\mathbb{C})$, $0 \\le s &lt; 1/2$, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of $L_u$.

Motivation & Objective

• To analyze the spectrum of the Lax operator $L_u = D - T_u$ for complex-valued potentials $u$ in $H^{-s}_{c,0}(\mathbb{T})$ with $0 \leq s < 1/2$.
• To extend the moment map $\Gamma(u) = (\gamma_n(u))_{n \geq 1}$, defined as the gap lengths of the spectrum, to a neighborhood of real $H^{-s}_{r,0}$ in complex $H^{-s}_{c,0}$.
• To establish the analyticity of the eigenvalue map $\lambda_n(u)$ and the moment map $\Gamma(u)$ in the complex Sobolev space $H^{-s}_{c,0}$.
• To extend the product representation of the generating function $\mathcal{H}_\lambda(u)$ to complex potentials via spectral data.

Proposed method

• Define the Lax operator $L_u = D - T_u$ on the Hardy space $H_+$, where $D = -i\partial_x$ and $T_u$ is the Toeplitz operator with potential $u$.
• Use the Szegő projector $\Pi$ to define $T_u f = \Pi(uf)$, ensuring $T_u$ maps $H_+$ to itself.
• Prove that $L_u$ has compact resolvent on $H^{1-s}_+$ for $u \in U^{-s} \subset H^{-s}_{c,0}$, implying discrete, simple eigenvalues $\lambda_n(u)$.
• Establish asymptotic behavior $|\lambda_n(u) - n| \to 0$ and bounds $|\lambda_{n+1}(u) - \lambda_n(u)| > 1/2$, $|\Im \lambda_n(u)| < 1/4$.
• Define the gap lengths $\gamma_n(u) = \lambda_n(u) - \lambda_{n-1}(u) - 1$ and prove $\Gamma(u) = (\gamma_n(u))_{n \geq 1} \in \ell^{1,1-2s}(\mathbb{N}, \mathbb{C})$.
• Use the notion of normally analytic maps to prove analyticity of $\lambda_n(u)$ and $\Gamma(u)$ in the complex domain $U^{-s}$.

Experimental results

Research questions

• RQ1 How does the spectrum of the Lax operator $L_u$ behave for complex-valued potentials $u \in H^{-s}_{c,0}$ with $0 \leq s < 1/2$?
• RQ2 Is the map $\Gamma(u) = (\gamma_n(u))_{n \geq 1}$, assigning gap lengths to potentials, analytic in a neighborhood of real $H^{-s}_{r,0}$?
• RQ3 Can the product representation of the generating function $\mathcal{H}_\lambda(u)$ be extended to complex potentials via spectral data?
• RQ4 What is the regularity of eigenvalues $\lambda_n(u)$ as functions of $u$ in the complex Sobolev space $H^{-s}_{c,0}$?
• RQ5 How do the spectral properties of $L_u$ relate to the integrability structure of the Benjamin-Ono equation on the torus?

Key findings

• For any $u \in U^{-s}$, the Lax operator $L_u$ has compact resolvent and a discrete spectrum consisting of simple eigenvalues $\lambda_n(u)$ with $\lim_{n \to \infty} |\lambda_n(u) - n| = 0$.
• The eigenvalues satisfy $|\lambda_{n+1}(u) - \lambda_n(u)| > 1/2$ and $|\Im \lambda_n(u)| < 1/4$ for all $n \geq 0$.
• The eigenvalue maps $\lambda_n: U^{-s} \to \mathbb{C}$ are analytic for each $n \geq 0$.
• The gap length map $\Gamma(u) = (\gamma_n(u))_{n \geq 1}$ with $\gamma_n(u) = \lambda_n(u) - \lambda_{n-1}(u) - 1$ takes values in $\ell^{1,1-2s}(\mathbb{N}, \mathbb{C})$.
• The map $\Gamma: U^{-s} \to \ell^{1,1-2s}(\mathbb{N}, \mathbb{C})$ is analytic, extending the moment map to complex potentials.
• The generating function $\mathcal{H}_\lambda(u) = \langle (L_u - \lambda)^{-1}1 | 1 \rangle$ admits a product representation for $u \in U^{-s}$, extending results from real potentials.