  We study a class of pseudo-differential operators with oscillating symbols or
osc illating amplitudes appearing in the long-range scattering theory. We
develop the basic calc ulus for operators from such classes and solve some
concrete problems posed by applications to scattering theory, especially to the
scattering matrix. In particular, we show tha t under natural assumptions the
spectrum of a pseudo-differential operator with an oscillating symb ol covers
the unit circle.

Spectral Theory;Analysis of PDEs