Diagonalizations of two classes of unbounded Hankel operators

We show that every Hankel operator $H$ is unitarily equivalent to a pseudo-differential operator $A$ of a special structure acting in the space $L^2(\mathbb{R})$. As an example, we consider integral operators $H$ in the space $L^2({\mathbb{R}}_{+})$ with kernels $P(\ln (t+s))(t+s{)}^{-1}$ where $P(x)$ is an arbitrary real polynomial of degree $K$. In this case, $A$ is a differential operator of the same order $K$. This allows us to study spectral properties of Hankel operators $H$ with such kernels. In particular, we show that the essential spectrum of $H$ coincides with the whole axis for $K$ odd, and it coincides with the positive half-axis for $K$ even. In the latter case we additionally find necessary and sufficient conditions for the positivity of $H$. We also consider Hankel operators whose kernels have a strong singularity at some positive point. We show that spectra of such operators consist of the zero eigenvalue of infinite multiplicity and eigenvalues accumulating to $+\mathrm{\infty }$ and $-\mathrm{\infty }$. We find the asymptotics of these eigenvalues.