Semiclassical quantization of non-Hermitian 2-D systems: Classical (Lie transform) Perturbation theory

Both real and complex semiclassical eigen energies of two dimensional non-Hermitian Hamiltonian systems are obtained by classical (Lie transform) perturbation theory requiring the action variables I 1 and I 2 to satisfy the quantization condition I 1 =(n 1 +(1/2))ħ and I 2 =(n 2 +(1/2))ħ respectively where n 1 , n 2 are integers. Classical perturbation theory with Lie transform makes classical trajectories, which are non-periodic or non-quasi-periodic, periodic. It was observed that this method produces accurate eigen energies even when classical trajectories are not periodic or quasi-periodic. Eigen energies obtained by classical perturbation theory are compared with the same, determined by Rayleigh-Schroedinger perturbation theory


INTRODUCTION
Real non-separable multidimensional Harmonic Oscillator Systems have been widely used in the past for investigating classical chaos and its manifestation in quantum mechanics in Hamiltonian systems 1-12 .Further more, semiclassical investigations of these oscillators provide insight into the various experiments related to the intramolecular energy transfer and understanding of the nature and meaning of intramolecular randomization¹³.Although non-Hermitian multidimensional Harmonic oscillator systems do not provide such a practical insight yet, semiclassical investigations of non-Hermitian complex Hamiltonian systems are important from both the fundamental and practical point of view.In the first place they may expose important clues hidden in the correspondence between classical and quantum mechanics, especially when classical motion becomes chaotic or PT symmetry of quantum states are spontaneously broken.Secondly, they provide powerful tools for calculations of highly excited energy spectra of the multidimensional complex systems.Particularly, when action variables or eigen energy expressions are available in analytic form, it is possible to have a better understanding on the classical behaviour, the semiclassical quantization and the semiclassical limit.
Classical perturbation theory based on Lie transforms has been used to obtain constants of motion of 2-D coupled Harmonic oscillator systems 14,15 and to determine semiclassical eigen energies and linking quantum avoided crossing with classical chaos in real Hermitian systems in the past 16,17 .Classical motion of 1-D non-Hermitian Hamiltonian systems has been studied in several investigations 1-21 .In this paper, we use Lie transforms to investigate the classical frequencies and real and complex semiclassical eigen values of 2-D non Hermitian Hamiltonian systems analytically.In particular, we study two Hamiltonians, H a and H b , which are non Hermitian and classical trajectories of H b which do not have periodic or quasi periodic trajectories, and obtained by solving the equation of motion in transformed coordinates.In section III, the semiclassical eigen values are calculated using classical perturbation theory and compares them with exact and Rayleigh-Schroedinger perturbation energies.Concluding remarks are given in section IV.

I Classical Lie canonical transform
In this section, first we describe the classical perturbation theory based on Lie canonical transformations.(For further details, see reference 14 & 15).Consider the Hamiltonian of the form where ] and H 1 (x 1 , x 2 ) makes the above Hamiltonian H non-separable and non-Hermitian.Hamilton's equation of motion for the above Hamiltonian is Now consider the transformation (p j , x j )→(η j , ξ j ) given by °¿ °¾ ½ °°® w w w w w w ... (7) where { } represents the Poisson bracket and equations of motion become  13) where the subscript s stands for the average value which is defined for a function A as (14) and for a periodic function with period T above equation becomes Now we obtain H 0 * , H * and equivalent constants of motion for two non-Hermitian systems given in equations ( 1) and (2).The Hamiltonian H a in (1) is PT symmetric and known to have all eigen values real.Assuming ω 1 is not commensurable with ω 2 , H 0 * (η,ξ) and H * (η,ξ) are found for the above system as > @ a a g H ... (16) where (17) This new system admits two first integrals, H a0 * =const and H a * =const and they are equivalent to const Above equations ( 16), ( 18) and ( 19) will be used to quantize this system in the section III.Now we find similar expressions of H 0 * and H * for the Hamiltonian systems H b . where The above Hamiltonians are functions of new variables ξ 1 , ξ 2 , η 1 , and η 2 only in the form ξ 1 ²+η 1 ²/ω 1 ² and ξ 2 ²+η 2 ²/ ω 2 ².These transformed Hamiltonians are merely the Borkhoff normal forms 1,22 of the original Hamiltonians.

II Classical frequencies
In the previous section we expressed the Hamiltonians of two non-Hermitian systems given by ( 1) and (2) in terms of constants of motion using classical Lie transform method.Constants of motion for these two dimensional potentials, are Therefore, functions of ξ 1 ²+η 1 ²/ω 1 ² and ξ 2 ²+η 2 ²/ω 2 ² are also constants of motion of those systems.In terms of c 1 and c 2 , Hamiltonians H a and H b become respectively.Using equation ( 8) and ( 9 where ϕ is an integration constant and It is important to note that Hamiltonians H k * (k={a,b}) are integrable, while corresponding original Hamiltonians H k are not.H k * are the integrable approximates to H k .It is evident from the above expressions for the frequencies of oscillations Ω k, j , that the classical trajectories of the first Hamiltonian system is periodic with real frequencies.However, the frequencies of the second Hamiltonian H b * are complex and classical trajectories of that Hamiltonian are therefore non-periodic.Under the Lie transform, non-integrable Hamiltonians became integrable and quasi-periodic trajectories of the first pseudo Hermitian Hamiltonian became periodic.However, trajectories of the second non-pseudo Hermitian Hamiltonians did not become periodic.Then an important question to address is that whether the semiclassical quantization of the above complex non-Hermitian systems accurately produce the quantum eigen values as it does in the case of real Hermitian systems.This issue is addressed in the next section.

III Semiclassical eigen values
In order to find semiclassical energy eigen values of those three Hamiltonians, first we make the transformation to action variables as or I j = (ξ j 2 + η j 2 / w j 2 ) w j π where and Now we quantize the systems H a and H b with the quantization conditions where n 1 and n 2 are the two quantum numbers.In this way, we can quantize both non-Hermitian Hamiltonians discussed above.Now we derive expressions for quantum mechanical eigen energies by applying aforementioned quantization conditions.

E
a n 1 n 2 and E b n 1 n 2 in (36) and (37) are the quantum eigen energies of the systems (1) and ( 2) respectively and n 1 and n 2 are non negative integers (quantum numbers).Tables 1 and 2 contain the eigen energies calculated using second order Lie transform method and the second order Rayleigh-Schroedinger perturbation theory.For comparison purposes, exact eigen energies which are obtained by diagonalizing the Hamiltonian in Harmonic oscillator basis set are also included in each table.The above two tables clearly show that the semiclassical eigen energies are in good agreement with exact quantum eigen values regardless of the Hermiticity of the Hamiltonians and whether the periods of the trajectories } } } are real or not.For the Hamiltonian H b , the second order Rayleigh-Schroedinger (RS) perturbation theory produces same complex energies as the semiclassical energies calculated with Lie transform method.

IV Concluding remarks
In this paper, we studied two 2D non Hermitian complex Hamiltonians, H a and H b with Classical Lie transform method.The first Hamiltonian H a is PT symmetric and the entire quantum energy spectrum is found to be real and positive.In this study it was found that classical phase space of H a contains periodic trajectories and analytic expression for frequency of oscillations was obtained.Semiclassical energies are obtained as an algebraic expression and semiclassical eigen energies are found to be in very good agreement with exact quantum eigen energies.
The second Hamiltonian H b is not even pseudo Hermitian.The entire quantum energy spectrum is complex.The classical frequencies are found to be complex and hence all the classical trajectories are non periodic.Since an action variable is intrinsically a concept associated with periodic motion, existence of action variable is directly linked with that of the periodic motion.However, semiclassical eigen energies based on quantization of the action variables in this system are found to be quite accurate although no trajectories of this system is periodic.
2) and the new Hamiltonian H * (η,ξ)=H(p,x).The purpose for introducing this canonical transformation is, to eliminate, from Hamiltonian H * , the parameter τ which is introduced in (11) and(12).Now we expand H * in a power series of ε as H* (η,ξ) = H 0 * (η,ξ) + εH 1 * (η,ξ ) + ... (10) 2) and hence H 0 * = const.The H k * s in the power series(10) are obtained (details are given in reference 14,15) by ... ( ), we write the equations of motion for the above systems in terms of c 1 and c 2 as 0 j are independent of time as c 1 and c 2 are constants of motion.Therefore, the solutions of equation motion (26 of the National Science Foundation of Sri Lanka 37(2) Therefore, Hamiltonians in terms of action variable I 1 and I 2 are given by