$J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach

TL;DR

This paper extends the self-adjoint extension method to construct $J$-self-adjoint Hamiltonians with complex point-interactions, using Krein space theory, and analyzes their $ ext{C}$-symmetries with applications to pseudo-Hermitian quantum models.

Contribution

It develops a framework for constructing and analyzing $J$-self-adjoint operators with complex interactions using extension theory and Krein space methods, including parametrizations of $ ext{C}$ operators.

Findings

Bijective correspondence between Hamiltonians and hypermaximal neutral subspaces.

Explicit $ ext{C}$-operator parametrizations for $<2,2>$ deficiency indices.

Application to 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians.

Abstract

A well known tool in conventional (von Neumann) quantum mechanics is the self-adjoint 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 ($J$-self-adjoint) 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 $<n,n>$. General properties of the $\cC$ operators for these Hamiltonians are derived. A detailed study of $\cC$-operator parametrizations and Krein type resolvent formulas is provided for $J$-self-adjoint extensions of symmetric operators with deficiency indices $<2,2>$. The technique is exemplified on 1D pseudo-Hermitian Schr\"odinger and Dirac Hamiltonians with complex point-interaction potentials.