Complex dynamical invariants for two-dimensional complex potentials (original) (raw)
Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems
Canadian_Journal_of_Physics
Keeping in view the importance of dynamical invariants, attempts have been made to investigate complex invariants for two-dimensional Hamiltonian systems within the framework of the extended complex phase space approach. The rationalization 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.
Complex invariants in two-dimension for coupled oscillator systems
In order to extract some insight into the features of a dynamical system, we present here the possibility of its complex dynamical invariant. There are many systems which admit complex invariants. To achieve this we use Lie algebraic method to study two dimensional complex systems (coupled oscillator system) on the extended complex phase plane characterized by x = x 1 + ip 3 , y = x 2 + ip 4 , px = p 1 + ix 3 , py = p 2 + ix 4. Such invariants play an important role in the analysis of complex trajectories with regard to the calculation of semi-classical coherent state propagator in the path integral method.
Search of exact invariants for PT and non-PT -symmetric complex Hamiltonian systems
Applied Mathematics and Computation
We build exact dynamical invariants corresponding to PT -symmetric (Parity and Time reversal) and non-PT -symmetric complex Hamiltonian systems in two dimensions, in order to obtain an additional insight into the features of dynamical Hamiltonian systems. There are many dynamical systems which admit complex invariants and simultaneously different methods are there to obtain it. The rationalization method is used to study two-dimensional complex dynamical systems on the extended complex phase plane. The role and scope of these invariants is pointed out. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c KAM theorem (dynamical systems about the persistence of quasi-periodic motions under small perturbations) [14], in classification of Yang-Mills Fields [15] etc.
Dynamical invariants for time-dependent real and complex Hamiltonian systems
Journal of Mathematical Physics
The Struckmeier and Riedel (SR) approach is extended in real space to isolate dynamical invariants for one-and two-dimensional timedependent Hamiltonian systems. We further develop the SR-formalism in zz complex phase space characterized by z = x + iy andz = x − iy and construct invariants for some physical systems. The obtained quadratic invariants contain a function f2(t), which is a solution of a linear third-order differential equation. We further explore this approach into extended complex phase space defined by x = x1 + ip2 and p = p1 + ix2 to construct a quadratic invariant for a time-dependent quadratic potential. The derived invariants may be of interest in the realm of numerical simulations of explicitly time-dependent Hamiltonian systems.
Invariant Eigen-Structure in Complex-Valued Quantum Mechanics
International Journal of Nonlinear Sciences and Numerical Simulation, 2009
The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle's complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the Schrodinger equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the complex-valued quantum mechanics in that the eigen-structures of real and complex quantum systems, such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant under linear complex mapping. In other words, there is indeed no distinction between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems when viewed from a complex domain. Their eigen-structures can be made coincident through linear transformation of complex coordinates.
Effects of complex parameters on classical trajectories of Hamiltonian systems
Pramana, 2014
Anderson et al have shown that for complex energies, the classical trajectories of real quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that, when time is complex t (t = t r e iθτ), certain real hermitian systems possess close periodic trajectories only for a discrete set of values of θ τ. On the other hand it is generally true that even for real energies, classical trajectories of non PTsymmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper we show that for given real energy, the classical trajectories of complex quartic Hamiltonians H = p 2 + ax 4 + bx k , (where a is real, b is complex and k = 1 or 2) are closed and periodic only for a discrete set of parameter curves in the complex b-plane. It was further found that given complex parameter b, the classical trajectories are periodic for a discrete set of real energies (i.e. classical energy get discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of complex Hamiltonian H = p 2 + µx 4 , (µ = µ r e iθ) are periodic when θ = 4tan −1 [(n/(2m + n))] for ∀ n and m ∈ Z.
Classical trajectories of 1D complex non-Hermitian Hamiltonian systems
Journal of Physics A: Mathematical and General, 2004
Classical motion of complex 1-D non-Hermitian Hamiltonian systems is investigated analytically to identify periodic, unbounded and chaotic trajectories. Expressions for Lyapunov exponent for 1-D complex Hamiltonians are derived. Complex potentials V 1 (x) = 1 2 µx 2 and V 2 (x) = µx 3 are studied in detail and their Lyapunov exponents are obtained analytically. It was found that when µ is complex all the trajectories of V1 are chaotic with Lyapunov exponent |Im(µ)| and most of the trajectories of V2 are periodic when µ is pure imaginary. But for other complex values of µ trajectories of V2 are non-periodic and show infinite oscillations. Unbounded neighbouring trajectories of V 2 show power law divergence rather than expoenential divergence as in the case of V1.
Physics Letters A, 2001
We propose that the real spectrum and the orthogonality of the states for several known complex potentials of both types, PT-symmetric and non-PT-symmetric can be understood in terms of currently proposed η-pseudo-Hermiticity (Mostafazadeh, quant-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, η, is introduced as e −θp which affects an imaginary shift of the coordinate : e −θp x e θp = x + iθ. Until year 1998 [1], Hermiticity of the Hamiltonian was supposed to be the necessary condition for having real spectrum. A conjecture due to Bender and Boettcher [1], has relaxed this condition in a very inspiring way by introducing the concept of PT-symmetric Hamiltonians. Here, P denotes the parity operation (space reflection) : x → −x and T mimics the timereversal : i → −i. Let χ denote PT then if (i)-χHχ −1 = H and if (ii)-χΨ(x) = ±1Ψ(x) the
Invariants and geometric phase for systems with non-Hermitian time-dependent Hamiltonians
Physical Review A, 1992
In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. It is then used to study the nonadiabatic cyclic evolution and the Aharonov-Anandan phase. It is shown that the study of noncyclic evolution can be reduced to the study of cyclic evolution. The two-level dissipative system and the classical time-dependent harmonic oscillator are discussed as illustrative examples.