J-self-adjoint operators with C-symmetries: extension theory approach
J-self-adjoint operators with C-symmetries: extension theory approach
Kuzhel, S.; Günther, U.; Albeverio, S.
Abstract
A linear densely defined operator A acting in a Krein space with fundamental symmetry J and indefinite metric [.,.]J =(J.,.) is called J-selfadjoint if A*J = JA. In contrast to self-adjoint operators in Hilbert spaces (which necessarily have a purely real spectrum), J-selfadjoint operators, in general, have a spectrum which is only symmetric with respect to the real axis. However, one can ensure the reality of the spectrum by imposing an extra condition of symmetry. In particular, a J-selfadjoint operator A has the property of C-symmetry if there exists a bounded linear operator C in H such that: (i) C2 =I; (ii) JC > 0; (iii) AC = CA.
The properties of C are nearly identical to those of the charge conjugation operator in quantum field theory and the existence of C provides an inner product (.,.)C=[C.,.]J whose associated norm is positive definite and the dynamics generated by A is therefore governed by a unitary time evolution. However, the operator C depends on the choice of A and its finding is a nontrivial problem.
The report deals with the construction of C-symmetries for J-selfadjoint extensions of a symmetric operator Asym with finite deficiency indices
The results are exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.
Keywords: J-selfadjoint operators; Krein space; extension theory; deficiency index; C-symmetry; Clifford algebra
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Lecture (Conference)
8th Workshop "Operator Theory in Krein Spaces and Inverse Problems", 18.-21.12.2008, Berlin, Deutschland
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