Geodesic
Expansion in eigenfunctions of a nonselfadjoint operator with purely continuous spectrum
Izvestiya. Mathematics, Tome 32 (1989) no. 1, pp. 113-139 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Expansion in eigenfunctions of a~nonselfadjoint operator with purely continuous spectrum},
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S. E. Cheremshantsev. Expansion in eigenfunctions of a nonselfadjoint operator with purely continuous spectrum. Izvestiya. Mathematics, Tome 32 (1989) no. 1, pp. 113-139. http://geodesic.mathdoc.fr/item/IM2_1989_32_1_a6/

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