Tuesday, October 23, 2012

1210.5709 (D. R. Yafaev)

Spectral and scattering theory for perturbations of the Carleman

D. R. Yafaev
We study spectral properties of the Carleman operator and, in particular, obtain an explicit formula for its resolvent. Then we consider perturbations of the Carleman operator $H_{0}$ by Hankel operators $V$ with kernels $v(t)$ decaying sufficiently rapidly as $t\to\infty$ and not too singular at $t=0$. Our goal is to develop scattering theory for pairs $H_{0}$, $H=H_{0} +V $ and to derive an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $H$. We also prove 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 $H_{0}$. We find simple conditions 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 obtain an explicit estimate of this number.
View original: http://arxiv.org/abs/1210.5709

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