J-self-adjoint operators with C-symmetries: extension theory approach

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 A_sym with finite deficiency indices . We present a general method allowing us: (i) to describe the set of J-selfadjoint extensions A of A_sym with C-symmetries; (ii) to construct the corresponding C-symmetries in a simple explicit form which is closely related to Clifford algebra operator structures; (iii) to establish a Krein-type resolvent formula for J-self adjoint extensions A with C-symmetries. 
The results are exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.