Spectral and scattering theory for differential and Hankel operators

Abstract

International audienceWe 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

Hankel and Toeplitz operators: continuous and discrete representations

“…The same arguments work if L is defined on the set F D + . Therefore Proposition 4.1 allows us to obtain a similar result for the operator H. Our goal is to obtain rather a complete information about the spectral structure of the closure of H which will also be denoted H. Given Proposition 4.1, this result is a consequence (except the assertion about the point 0 which is proven in Theorem 4.7 of [16]) of the corresponding statement, Theorem 4.8 in [21], for the differential operator L. We emphasize that the method of [21] yields a sufficiently explicit spectral analysis of the operator L and, in particular, information about its eigenfunctions of the continuous spectrum. In view of the unitary equivalence, this yields the corresponding results for the Hankel operator H (and H = LHL * ), but we will not dwell upon them.…”

“…We exhibit two quite different cases where the spectral analysis of Hankel operators H can be carried out sufficiently explicitly. Our approach relies on Theorem 3.3 and the results on Hankel operators H with singular integral kernels a(t) obtained earlier in [18,21].…”

“…It turns out that even this particular case leads to interesting examples. Our plan is to use the results obtained in [18,21] for integral Hankel operators H, and then with the help of Theorem 3.6 to state new results for matrix Hankel operators H.…”

“…Our study of the Hankel operator H with matrix elements (4.5) relies on a combination of Theorem 3.6 with the results of [16], [21] on integral Hankel operators H in the space L 2 (R + ). In view of (4.3) the sigma-function of H = LHL * equals σ(λ) = p l=0 γ l ln l λ, p ≥ 1, and its integral kernel a(t) is given by formula (4.1).…”

A new representation of Hankel operators and its spectral consequences

“…Thus, even in this relatively simple question, the degeneracy of v(x) at infinity significantly changes spectral properties of differential operators A. The detailed spectral structure, in particular, the absolutely continuous spectrum, of differential operators (1.10) and hence of the Hankel operators with kernels (1.7) was described in [20].…”