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Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential. (English) Zbl 1124.47029
Summary: We study dissipative Schrödinger operators with a matrix potential in ${L}_{2}\left(\left(0,\infty \right);E\right)$, $dimE=n<\infty$), which are the extension of a minimal symmetric operator ${L}_{0}$ with defect index $\left(n,n\right)$. A selfadjoint dilation of the dissipative operator is constructed and, using the Lax–Phillips scattering theory, the spectral analysis is carried out and the scattering matrix is found. A functional model of the dissipative operator is constructed, the analytic properties of its characteristic function are determined, and some theorems on the completeness of eigenvectors and associated vectors of dissipative Schrödinger operators are proved.
##### MSC:
 47E05 Ordinary differential operators 34L40 Particular ordinary differential operators 47B25 Symmetric and selfadjoint operators (unbounded) 47B44 Accretive operators, dissipative operators, etc. (linear) 47N50 Applications of operator theory in quantum physics