Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems J.S. Virdi, F. Chand, C.N. Kumar, and S.C. Mishra Abstract: Keeping in view the importance of dynamical invariants, attempts have been made to investigate complex invari- ants for two-dimensional Hamiltonian systems within the framework of the extended complex phase space approach. The ra- tionalization method has been used to derive an invariant of a general nonhermitian quartic potential. Invariants for three specific potentials are also obtained from the general result. PACS Nos: 02.30.IK, 03.65.Fd, 03.20+i Résumé : Connaissant l’importance des invariants dynamiques, nous avons essayé d’étudier les invariants complexes de sys- tèmes hamiltoniens en deux dimensions, dans une approche utilisant l’espace de phase complexe étendu. Nous avons utilisé une méthode de rationalisation pour obtenir un invariant d’un potentiel quartique général non hermitien. À partir de ce cas général, nous trouvons des invariants pour trois potentiels spécifiques. [Traduit par la Rédaction] 1. Introduction The theory of dynamical invariants has been applied in a variety of fields like plasma physics, laser physics, hydrody- namics, astrophysics, and accelerator physics to explore the underlying dynamics of the concerned systems [1]. In the past, several attempts have been made to construct invariants in different forms for a variety of systems and to find possi- ble applications [2–5], but most of such studies are confined to one-dimensional systems. Therefore, the study of invari- ants for higher dimensional systems is pertinent from the point of view of both mathematical interest and realistic ap- plications. It is well known that the Hamiltonian representation of a dynamical system in real space provides a lot of information about the system. But sometimes, a complex form of the Hamiltonian for a system may provide some additional infor- mation. In the past, complex potentials were generally used to model dissipative systems, such as the optical model of the nucleus [6]. Thus complex potentials deserve a detailed classical and quantum mechanical study. After publication of the seminal paper by Bender and Boettcher in 1998 [7], quantum mechanics of nonhermitian complex potentials, particularly PT-symmetric Hamiltonians, has been widely studied [8]. But little effort has been made on the classical front. So construction of classical invariants for nonhermitian systems can be important. Recently, Kau- shal and his co-workers [9–11] used an extended complex phase space (ECPS), characterized by x = x 1 + ip 2 and p = p 1 + ix 2 to study some classical and quantum aspects of non- hermitian Hamiltonians. Similar transformations have also been used in some other studies [12–14]. The ECPS ap- proach was further used in refs. 15 and 16 for the study of two-dimensional nonhermitian quantum systems. As far as the construction of invariants for nonhermitian systems in an ECPS is concerned, only a few studies of one- dimensional systems have been reported in the recent past [9, 11]. Therefore, in the present study, we obtain invariants for a general nonhermitian quartic potential in two-dimensional ECPS. Although many methods for the construction of real and (or) complex invariants may be found in the literature, here we follow the rationalization method that has been widely used in the past [3–5, 17]. This method is quite straightfor- ward and can provide exact invariants. The organization of the paper is as follows. In Sect. 2, a brief description of the rationalization method for two- dimensional nonhermitian systems is presented. Section 3 comprises the details of construction of invariants for a gen- eral quartic potential and its three variants. Finally, conclud- ing remarks are given in Sect. 4. 2. Formalism Consider a two-dimensional real phase space (x, y, p x , p y , t) that may be transformed into an ECPS (x 1 , p 3 , x 2 , p 4 , p 1 , x 3 , p 2 , x 4 , t) by defining variables for position and momen- tum degrees of freedom as x ¼ x 1 þ ip 3 y ¼ x 2 þ ip 4 p x ¼ p 1 þ ix 3 p y ¼ p 2 þ ix 4 ð1Þ The presence of variables (p 3 , p 4 , x 3 , x 4 ) in transformations (1) may be regarded as some sort of coordinates–momenta Received 6 August 2011. Accepted 30 November 2011. Published at www.nrcresearchpress.com/cjp on 31 January 2012. J.S. Virdi and C.N. Kumar. Department of Physics, Panjab University, Chandigarh 160014, India. F. Chand and S.C. Mishra. Department of Physics, Kurukshetra University, Kurukshetra 136119, India. Corresponding authors: J.S. Virdi (e-mail: jpsvirdi@gmail.com) and F. Chand (e-mail: fchand@kuk.ac.in). 151 Can. J. Phys. 90: 151–157 (2012) doi:10.1139/P11-152 Published by NRC Research Press Can. J. Phys. Downloaded from www.nrcresearchpress.com by Dalian Nationalities University on 06/07/13 For personal use only.