[Paper Review] On the eigenvalues of some nonhermitian oscillators

Francisco M. Fern, andez+1 more|arXiv (Cornell University)|Jan 7, 2013

Quantum Mechanics and Non-Hermitian Physics1 references3 citations

TL;DR

This paper investigates the eigenvalues and resonances of non-Hermitian oscillators, particularly PT-symmetric systems, using complex rotation and Riccati-Padé methods. It demonstrates that the optimal rotation angle transforms the system into either a PT-symmetric or Hermitian form, and shows that both real positive eigenvalues and resonances can be computed via diagonalization and Riccati-Padé techniques, with the latter method capturing both types of states.

ABSTRACT

We consider a class of one-dimensional nonhermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. Comparison of the results of the complex rotation and the Riccati-Pad\'{e} methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or an Hermitian one. In addition to the real positive eigenvalues the PT-symmetric oscillators exhibit real positive resonances under different boundary conditions. They can be calculated by means of the straightforward diagonalization method. The Riccati-Pad\'e method yields not only the resonances of the nonhermitian oscillators but also the eigenvalues of the PT-symmetric ones.

Motivation & Objective

• To clarify the relationship between real eigenvalues in PT-symmetric oscillators and resonances obtained by other authors.
• To analyze strong-coupling expansions of eigenvalues for non-Hermitian oscillators.
• To investigate how the optimal rotation angle transforms non-Hermitian oscillators into either PT-symmetric or Hermitian forms.
• To demonstrate that both resonances and eigenvalues of PT-symmetric oscillators can be computed using the Riccati-Padé method.

Proposed method

• Application of the complex rotation method to transform the Hamiltonian and extract resonances.
• Employment of the Riccati-Padé method to compute eigenvalues and resonances of non-Hermitian oscillators.
• Comparison of results from complex rotation and Riccati-Padé methods to identify optimal rotation angles.
• Use of straightforward diagonalization to calculate real positive resonances under different boundary conditions.
• Analysis of strong-coupling expansions to relate eigenvalue behavior across different oscillator models.
• Identification of conditions under which the optimal rotation angle leads to PT-symmetric or Hermitian Hamiltonians.

Experimental results

Research questions

• RQ1 How do the real eigenvalues of PT-symmetric oscillators relate to the resonances found by other authors?
• RQ2 What is the role of the optimal rotation angle in transforming non-Hermitian oscillators into PT-symmetric or Hermitian forms?
• RQ3 How do the results of the complex rotation and Riccati-Padé methods compare in computing resonances and eigenvalues?
• RQ4 Can the Riccati-Padé method accurately compute both resonances and eigenvalues in non-Hermitian and PT-symmetric oscillators?
• RQ5 What is the behavior of strong-coupling expansions for the eigenvalues of non-Hermitian oscillators?

Key findings

• The optimal rotation angle transforms the non-Hermitian oscillator into either a PT-symmetric or Hermitian system, depending on the parameter regime.
• Real positive resonances emerge in PT-symmetric oscillators under different boundary conditions and can be computed via straightforward diagonalization.
• The Riccati-Padé method successfully computes both resonances of non-Hermitian oscillators and eigenvalues of PT-symmetric ones.
• Strong-coupling expansions of eigenvalues for non-Hermitian oscillators are consistently related across different models.
• Comparison between complex rotation and Riccati-Padé methods confirms the robustness of the optimal rotation angle in simplifying the system.
• The study establishes a direct link between resonances in non-Hermitian systems and eigenvalues in PT-symmetric systems through shared mathematical structure.