Selfadjoint dilatation of the dissipative Shr\"odinger operator and its resolution in terms of eigenfunctions

B. S. Pavlov

Sbornik. Mathematics, Tome 31 (1977) no. 4, pp. 457-478

Résumé

The object of the present work is the imbedding of the spectral theory for the dissipative Schrödinger operator $L$ with absolutely continuous spectrum acting in the Hilbert space $H=L_2(R^3)$ in the spectral theory of a model operator and the proof of the theorem on expansion in terms of eigenfunctions. The imbedding mentioned is achieved by constructing a selfadjoint dilation $\mathscr L$ of the operator $L$. In the so-called incoming spectral representation of this dilation the operator becomes the corresponding model operator. Next, a system of eigenfunctions of the dilation – the “radiating” eigenfunctions – is constructed. From these a canonical system of eigenfunctions for the absolutely continuous spectrum of the operator and its spectral projections are obtained by “orthogonal projection” onto $H$. Bibliography: 22 titles.