Spectral and scattering theory for differential and Hankel operators

TL;DR

This paper develops a spectral theory framework for a class of Hankel operators with polynomial kernels, reducing them to differential operators, and characterizes their spectral properties including the nature of their spectra.

Contribution

It introduces an explicit unitary reduction of Hankel operators with polynomial kernels to differential operators and analyzes their spectral properties, including spectrum type and multiplicity.

Findings

Absolutely continuous spectrum is simple and equals ℝ for odd n.

For even n ≥ 2, the spectrum has multiplicity 2 and equals [0, ∞).

Singular continuous spectrum is empty; eigenvalues may accumulate at zero.

Abstract

We consider a class of Hankel operators $H$ realized in the space $L^2 ({\Bbb R}_{+}) $ as integral operators with kernels $h(t+s)$ where $h(t)=P (\ln t) t ^{-1}$ and $P(X)= X^n+p_{n-1} X^{n-1}+\cdots$ is an arbitrary real polynomial of degree $n$. This class contains the classical Carleman operator when $n =0$. We show that a Hankel operator $H$ in this class can be reduced by an {\it explicit} unitary transformation (essentially by the Mellin transform) to a differential operator $A = v Q(D) v$ in the space $L^2 ({\Bbb R}) $. Here $Q(X)= X^n+ q_{n-1} X^{n-1}+\cdots$ is a polynomial determined by $P(X)$ and $v(\xi)=\pi^{1/2} (\cosh(\pi\xi))^{-1/2} $ is the universal function. Then the operator $A = v Q(D) v$ reduces by the generalized Liouville transform to the standard differential operator $B = D^n+ b_{n-1} (x)D^{n-1}+\cdots+ b_{0} (x)$ with the coefficients $b_{m}(x)$, $m=0,\ldots, n-1$, decaying sufficiently rapidly as $|x|\to \infty$. This allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized Carleman operators. In particular, we show that the absolutely continuous spectrum of $H$ is simple and coincides with $\Bbb R$ if $n$ is odd, and it has multiplicity $2$ and coincides with $[0,\infty)$ if $n\geq 2$ is even. The singular continuous spectrum of $H$ is empty, and its eigenvalues may accumulate to the point $0$ only. As a by-product of our considerations, we develop spectral theory of a new class of {\it degenerate} differential operators $A = v Q(D) v$ where $Q(X)$ is an arbitrary real polynomial and $v(\xi)$ is a sufficiently arbitrary real function decaying at infinity.