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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.

47E05Ordinary differential operators
34L40Particular ordinary differential operators
47A20Dilations, extensions and compressions of linear operators
47A45Canonical models for contractions and nonselfadjoint operators
47B25Symmetric and selfadjoint operators (unbounded)
47B44Accretive operators, dissipative operators, etc. (linear)
Full Text: DOI
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