Expansion in eigenfunctions of a~nonselfadjoint operator with purely continuous spectrum

S. E. Cheremshantsev

Izvestiya. Mathematics , Tome 32 (1989) no. 1, pp. 113-139

Résumé

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.