Investigation of PT-symmetric Hamiltonian systems from an alternative point of view

TL;DR

This paper explores PT-symmetric Hamiltonian systems using an algebraic approach that preserves the Hilbert space, introduces a new operator V for positive inner products, and confirms spectral properties consistent with existing literature.

Contribution

It applies an algebraic method to PT-symmetric systems, introducing a new operator V for inner products, and directly constructs it from Hamiltonians, offering a novel perspective.

Findings

Spectra match existing literature results

Hilbert spaces with positive definite inner products are constructed

Operator V can be directly built from Hamiltonians

Abstract

Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.