On $J$-self-adjoint extensions of the Phillips symmetric operator

Details On $J$-self-adjoint extensions of the Phillips symmetric operator On $J$-self-adjoint extensions of the Phillips symmetric operator

2009-07-19

arxiv.org/abs/0907.3280v2

Methods Funct. Anal. Topology 16 (2010), no. 4, 333-348

Physics

Mathematical Physics

Scientific paper

$J$-self-adjoint extensions of the Phillips symmetric operator $S$ are studied. The concepts of stable and unstable $C$-symmetry are introduced in the extension theory framework. The main results are the following: if ${A}$ is a $J$-self-adjoint extension of $S$, then either $\sigma({A})=\mathbb{R}$ or $\sigma({A})=\mathbb{C}$; if ${A}$ has a real spectrum, then ${A}$ has a stable $C$-symmetry and ${A}$ is similar to a self-adjoint operator; there are no $J$-self-adjoint extensions of the Phillips operator with unstable $C$-symmetry.

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