Expansion in eigenfunctions of a nonselfadjoint operator with purely continuous spectrum

The differential operator
$$ H=-\Delta_{\boldsymbol x}+i\varkappa\Delta_{\boldsymbol y}+q(\boldsymbol x-\boldsymbol y), $$
arising in the three-dimensional problem of scattering by a Brownian particle is studied. Its analysis reduces to the investigation of a family of operators in $L_2(\mathbf R^3)$:
$$ B_{\boldsymbol p}=-\Delta_{\boldsymbol v}+2(\boldsymbol p,\Delta_{\boldsymbol v})+\frac{q(\boldsymbol v)}{1-i\varkappa}, \quad \boldsymbol p\in \mathbf R^3. $$
Under the condition that the potential $q$ is bounded and small, an expansion in the eigenfunctions of the continuous spectrum of $B_\boldsymbol p$ is obtained. From this expansion an explicit formula is found for the semigroup $\exp(itH)$ on a set dense in $L_2(\mathbf R^6)$.
Bibliography: 5 titles.