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Register a new userA solvable dilation model of PT -symmetric systems
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The dilation method is an important and useful way in experimentally simulating non-Hermitian, especially $cal PT$-symmetric systems. However, the time dependent dilation problem cannot be explicitly solved in general. In this paper, we consider a special two dimensional time dependent $cal PT$-symmetric system, which is initially set in the unbroken $cal PT$-symmetric phase and later goes across the exceptional point and enters the broken $cal PT$-symmetric phase. For this system, the dilation Hamiltonian and the evolution of $cal PT$-symmetric system are analytically worked out.
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Three ways of constructing a non-Hermitian matrix with possible all real eigenvalues are discussed. They are PT symmetry, pseudo-Hermiticity, and generalized PT symmetry. Parameter counting is provided for each class. All three classes of matrices have more real parameters than a Hermitian matrix with the same dimension. The generalized PT-symmetric matrices are most general among the three. All self-adjoint matrices process a generalized PT symmetry. For a given matrix, it can be both PT-symmetric and P-pseudo-Hermitian with respect to some P operators. The relation between corresponding P and P operators is established. The Jordan block structures of each class are discussed. Explicit examples in 2x2 are shown.
This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: $V=-W^2+iW_x$. We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified by equations with attractive and repulsive cubic nonlinearity bearing a variety of bright and dark solitons.
Non-hermitian, $mathcal{PT}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, $mathcal{PT}$ potentials for a non-relativistic particle confined in a circular geometry. We show that the $mathcal{PT}$ symmetry threshold can be tuned by introducing a second gain-loss potential or its hermitian counterpart. Our results explicitly demonstrate that $mathcal{PT}$ breaking in two dimensions has a rich phase diagram, with multiple re-entrant $mathcal{PT}$ symmetric phases.
We introduce the simplest one-dimensional nonlinear model with the parity-time (PT) symmetry, which makes it possible to find exact analytical solutions for localized modes (solitons). The PT-symmetric element is represented by a point-like (delta-functional) gain-loss dipole {delta}^{prime}(x), combined with the usual attractive potential {delta}(x). The nonlinearity is represented by self-focusing (SF) or self-defocusing (SDF) Kerr terms, both spatially uniform and localized ones. The system can be implemented in planar optical waveguides. For the sake of comparison, also introduced is a model with separated {delta}-functional gain and loss, embedded into the linear medium and combined with the {delta}-localized Kerr nonlinearity and attractive potential. Full analytical solutions for pinned modes are found in both models. The exact solutions are compared with numerical counterparts, which are obtained in the gain-loss-dipole model with the {delta}^{prime}- and {delta}- functions replaced by their Lorentzian regularization. With the increase of the dipoles strength, {gamma}, the single-peak shape of the numerically found mode, supported by the uniform SF nonlinearity, transforms into a double-peak one. This transition coincides with the onset of the escape instability of the pinned soliton. In the case of the SDF uniform nonlinearity, the pinned modes are stable, keeping the single-peak shape.
We consider the linear and nonlinear Schrodinger equation for a Bose-Einstein condensate in a harmonic trap with $cal {PT}$-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent proposition by Mityagin and Siegl on singular perturbations of harmonic oscillator type self-adjoint operators are fulfilled. In both the linear and nonlinear case we calculate numerically the shifts of the unperturbed levels with quantum numbers $n$ of up to 89 in dependence on the strength of the non-Hermiticity and compare with rigorous estimates derived by those authors. We confirm that the predicted $1/n^{1/2}$ estimate provides a valid upper bound on the the shrink rate of the numerical eigenvalues. Moreover, we find that a more recent estimate of $log(n)/n^{3/2}$ is in excellent agreement with the numerical results. With nonlinearity the shrink rates are found to be smaller than without nonlinearity, and the rigorous estimates, derived only for the linear case, are no longer applicable.
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