On the spectral properties of the Hilbert transform operator on multi-intervals

TL;DR

This paper investigates the spectral properties of an operator related to the Hilbert transform on multi-intervals, revealing conditions for continuous spectrum and eigenvalues, using Riemann-Hilbert problem techniques.

Contribution

It introduces a self-adjoint operator $ ext{K}$ and analyzes its spectrum, including the resolvent construction via Riemann-Hilbert problems, providing new insights into spectral behavior on multi-intervals.

Findings

Spectrum has an absolutely continuous part [0,1] if and only if $J$ and $E$ share endpoints.

Eigenvalues may exist inside the spectrum, with finite multiplicity and possible accumulation at zero.

No singular continuous spectrum is present in all cases.

Abstract

Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1\pi\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\mathscr R$ in the spectral parameter $\lambda$. We show that the spectrum of $\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\dagger A$ and $A A^\dagger$, which are very similar to those of $\mathscr K$, are obtained as well.