Spectral asymptotics for compact self-adjoint Hankel operators

TL;DR

This paper investigates the eigenvalue asymptotics of compact self-adjoint Hankel operators, providing explicit formulas and revealing fundamental principles that govern their spectral behavior.

Contribution

It introduces the localization and symmetry principles, offering a comprehensive description of eigenvalue asymptotics for broad classes of Hankel operators.

Findings

Eigenvalues exhibit power asymptotics with explicit coefficients.

Disjoint singular support components contribute independently to eigenvalue asymptotics.

Spectral symmetry arises when the symbol's singular support excludes certain critical points.

Abstract

We describe large classes of compact self-adjoint Hankel operators whose eigenvalues have power asymptotics and obtain explicit expressions for the coefficient in front of the leading term. The results are stated both in the discrete and continuous representations for Hankel operators. We also elucidate two key principles underpinning the proof of such asymptotic relations. We call them {\it the localization principle} and {\it the symmetry principle}. The localization principle says that disjoint components of the singular support of the symbol of a Hankel operator make independent contributions into the asymptotics of eigenvalues. The symmetry principle says that if the singular support of a symbol does not contain the points $1$ and $-1$ in the discrete case (or the points $0$ and $\infty$ in the continuous case), then the spectrum of the corresponding Hankel operator is asymptotically symmetric with respect to the reflection around zero.