Spectral and scattering theory for perturbations of the Carleman operator

D. R. Yafaev

Algebra i analiz, Tome 25 (2013) no. 2, pp. 251-278.

Résumé

The spectral properties of the Carleman operator (the Hankel operator with the kernel $h_0(t)=t^{-1}$) are studied; in particular, an explicit formula for its resolvent is found. Then, perturbations are considered of the Carleman operator $H_0$ by Hankel operators $V$ with kernels $v(t)$ decaying sufficiently rapidly as $t\to \mathrm{\infty }$ and not too singular at $t=0$. The goal is to develop scattering theory for the pair $H_0$, $H=H_0+V$ and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $H$. Also, it is proved that, under general assumptions, the singular continuous spectrum of the operator $H$ is empty and that its eigenvalues may accumulate only to the edge points $0$ and $\pi$ in the spectrum of $H_0$. Simple conditions are found for the finiteness of the total number of eigenvalues of the operator $H$ lying above the (continuous) spectrum of the Carleman operator $H_0$, and an explicit estimate of this number is obtained. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.

Détail

Mots-clés : Hankel operators, resolvent kernels, absolutely continuous spectrum, eigenfunctions, wave operators, scattering matrix, resonances, discrete spectrum, total number of eigenvalues.

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