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On Integrability and Pseudo-Hermitian Systems Spin-Coupling Point Interactions

Shao-Ming Fei

Published Jan 1, 2005• 5 citations
OpenAlex

Abstract

We study the pseudo-Hermitian systems with general spin-coupling point interactions and give a systematic description of the corresponding boundary conditions for PT-symmetric systems. The corresponding integrability for both bosonic and fermionic many-body systems with PT-symmetric contact interactions is investigated.

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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.

arXiv:1204.6544•Communications in Theoretical Physics•Oct 1, 2012•13 citations
We study the quantum entanglement and separability of Hermitian and pseudo-Hermitian systems of identical bosonic or fermionic particles with point interactions. The separability conditions are investigated in detail. PACS: 02.30.Ik, 11.30.Er, 03.65.Fd Key words: Point interactions, PT-symmetry, Separability Pseudo-Hermitian quantum mechanical systems have been extensively investi-gated due to some mathematical and physical considerations [1]. For PT-symmetric systems with singular contact interactions, we have studied the classification, spec-tra and integrability problem in [2]. The results are generalized to the case of identical bosonic and fermionic many-body systems with PT-symmetric contact interactions and spin-coupling interactions [3]. On the other hand, as one of the most striking features of quantum phenom-ena [4], quantum entanglement has been identified as a key non-local resource in quantum information processing such as quantum computation [5], quantum tele-portation, dense coding, quantum cryptographic schemes, entanglement swapping and remote state preparation (RSP). Nevertheless the theory of quantum entan-glement is far from being satisfied. One of main problem in quantum entanglement is to judge a quantum state is separable or entangled, i.e. the separability. In this paper we study separability of Hermitian and pseudo-Hermitian systems of many body systems with contact interactions. We first consider contact interactions without spin coupling. A particle moving in one dimension with point interactions at the origin can be characterized by the non-separated1) boundary conditions imposed on the wave function ϕ at x = 0, ϕ
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.