$J$-self-adjoint operators with $mathcal{C}$-symmetries: extension theory approach - Mathematical PhysicsReport as inadecuate

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Abstract: A well known tool in conventional von Neumann quantum mechanics is theself-adjoint extension technique for symmetric operators. It is used, e.g., forthe construction of Dirac-Hermitian Hamiltonians with point-interactionpotentials. Here we reshape this technique to allow for the construction ofpseudo-Hermitian $J$-self-adjoint Hamiltonians with complexpoint-interactions. We demonstrate that the resulting Hamiltonians arebijectively related with so called hypermaximal neutral subspaces of the defectKrein space of the symmetric operator. This symmetric operator is allowed tohave arbitrary but equal deficiency indices $$. General properties of the$\cC$ operators for these Hamiltonians are derived. A detailed study of$\cC$-operator parametrizations and Krein type resolvent formulas is providedfor $J$-self-adjoint extensions of symmetric operators with deficiency indices$<2,2>$. The technique is exemplified on 1D pseudo-Hermitian Schr\-odinger andDirac Hamiltonians with complex point-interaction potentials.

Author: S. Albeverio, U. Guenther, S. Kuzhel

Source: https://arxiv.org/


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