Coupled oscillator systems having partialPTsymmetry

Abstract

This paper examines chains of N coupled harmonic oscillators. In isolation, the jth oscillator (1 ≤ j ≤ N ) has the natural frequency ω j and is described by the HamiltonianThe oscillators are coupled adjacently with coupling constants that are purely imaginary; the coupling of the jth oscillator to the (j + 1)st oscillator has the bilinear form iγx j x j+1 (γ real). The complex Hamiltonians for these systems exhibit partial PT symmetry; that is, they are invariant under i → −i (time reversal), x j → −x j (j odd), and x j → x j (j even). [They are also invariant under i → −i, x j → x j (j odd), and x j → −x j (j even).] For all N the quantum energy levels of these systems are calculated exactly and it is shown that the ground-state energy is real. When ω j = 1 for all j, the full spectrum consists of a real energy spectrum embedded in a complex one; the eigenfunctions corresponding to real energy levels exhibit partial PT symmetry. However, if the ω j are allowed to vary away from unity, one can induce a phase transition at which all energies become real. For the special case N = 2, when the spectrum is real, the associated classical system has localized, almost-periodic orbits in phase space and the classical particle is confined in the complexcoordinate plane. However, when the spectrum of the quantum system is partially real, the corresponding classical system displays only open trajectories for which the classical particle spirals off to infinity. Similar behavior is observed when N > 2.

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

“…As discussed in [12][13][14][15][16][17][18], both of partially PT symmetric potentials and PT symmetric potentials in multidimensions have an important role in optics, our results may have interesting applications in optics. Moreover, it is interesting to reduce the above solutions to other (1+1)-dimensional nonlocal equations, besides the nonlocal NLS equation.…”

“…Recently, the concept of PT symmetry in multidimensions has been generalized to include partialparity-time symmetry, and it is shown that partial-parity-time-symmetric systems share most of the properties of PT systems [12]. Besides, the partially PT symmetry has been shown possible interesting applications in optics [12][13][14][15][16][17][18]. Thus, more researches about partially PT symmetry are also inevitable and worthwhile, such as the integrable models and nonlinear waves in this system.…”

“…Motivated by the physical importance of partially and fully PT -symmetry systems in the multidimension space [12][13][14][15][16][17][18], we investigate above two types of the nonlocal DS equations. In this work, we focus on following aspects:…”

Analytic eigenvalue structure of a coupled-oscillator system beyond the ground state

“…For the calculation of the general energy function E 1 (g) we can thus focus on the system of remaining equations (17) to (22). This system again contains one part that describes the energy eigenvalues of the system and one that relates the parameters α, β and γ to the natural oscillator frequencies ν and ω and the coupling constant g. The two coupled energy equations (17) and (18) can be combined to yield the quadratic energy equation…”

Exactly solvable $\mathcal{PT}$ -symmetric models in two dimensions

“…In two or higher dimensions, the spectrum of the H 0 is degenerate for both lattice and continuum models. A generic PT -symmetric potential V connects these degenerate states and thus leads to a complex energy spectrum at arbitrarily small strengths of V [44,45]. A group theoretical analysis of these degeneracies and their effect on the PT -breaking transition threshold was carried out by Ge and Stone [46].…”