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J-selfadjoint operators with C-symmetries: extension theory approach
Albeverio, S.; Günther, U.; Kuzhel, S.;
A well known tool in conventional (von Neumann) quantum mechanics is the selfadjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac-Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian (J-selfadjoint) Hamiltonians with complex point-interactions. We demonstrate that the resulting Hamiltonians are bijectively related with so called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices . General properties of the C-operators for these Hamiltonians are derived. A detailed study of C-operator parametrizations and Krein type resolvent formulas is provided for J-selfadjoint extensions of symmetric operators with deficiency indices <2,2>. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.
Keywords: PT-symmetric quantum mechanics, pseudo-Hermitian operators, Krein space, extension theory, point interactions, hypermaximal neutral subspace, C-operator, super-symmetry, contraction mapping, resolvent, defect index, defect subspace, extension center

Publ.-Id: 11882 - Permalink