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Spectral analysis of dissipative Dirac operators with general boundary conditions. (English) Zbl 1093.47045
The author of this paper deals with a $2 \times 2$ symmetric system of the differential operators of Dirac type $A(x)^{-1}[J \frac{d}{dx} + B(x)]$ in $\Bbb R$, where $J= \left( \smallmatrix 0& -1 \\ 1& 0 \endsmallmatrix \right)$, and $A(x) >0$ and $B(x)$ are $2 \times 2$ Hermitian matrices with elements being real-valued, locally integrable functions in $\Bbb R$. It is considered as an operator in the Hilbert space $L_A(\Bbb R; \Bbb C^2)$ of the $\Bbb C^2$-valued measurable functions in $\Bbb R$ with the inner product $(y,z) = \int_{\Bbb R}(A(x) y(x), z(x))_{\Bbb C^2}\, dx$. The main purpose of the present paper is, for the corresponding minimal operator with deficiency indices $(2,2)$, to investigate all its maximal dissipative, selfadjoint, and/or other extensions in terms of boundary conditions at infinity. The dilation theory is also employed.

##### MSC:
 47E05 Ordinary differential operators 34L40 Particular ordinary differential operators 47A20 Dilations, extensions and compressions of linear operators 47A45 Canonical models for contractions and nonselfadjoint operators 47B25 Symmetric and selfadjoint operators (unbounded) 47B44 Accretive operators, dissipative operators, etc. (linear)
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