Related papers: Integrable and Superintegrable Klein-Gordon and Sc…
An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that…
Brief review is given of my recent results on solvable models within the so called PT symmetric version of quantum mechanics.
The spectral and localization properties of $\mathcal{PT}$-symmetric optical superlattices, either infinitely extended or truncated at one side, are theoretically investigated, and the criteria that ensure a real energy spectrum are…
$\mathcal{PT}$-symmetry --- invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$ --- provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general…
Supersymmetry between bosons and fermions is modeled within PT- symmetric quantum mechanics. A non-Hermitian alternative to the Witten's supersymmetric quantum mechanics is obtained.
Recent progress on nonlinear properties of parity-time ($\cal PT$-) symmetric systems is comprehensively reviewed in this article. $\cal PT$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\cal PT$…
In this paper, we consider exponential integrators for semilinear Poisson systems. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for…
Parity-time-symmetric ($\mathcal{PT}$-symmetric) optical waveguide couplers offer a great potential for future applications in integrated optics. Studies of nonlinear $\mathcal{PT}$-symmetric couplers present new possibilities for…
A real potential Hamiltonian has real energy bound states below the scattering threshold and complex energy resonances above it. Scattering states are not square integrable, being instead delta function normalized. This lack of square…
The pseudo-perturbation shifted-l expansion technique PSLET is shown applicable in the non-Hermitian PT-symmetric context. The construction of bound states for several PT-symmetric potentials is presented, with special attention paid to…
We show that the parity-time (PT) symmetric coupled optical waveguides with gain and loss support localised oscillatory structures similar to the breathers of the classical $\phi^4$ model. The power carried by the PT-breather oscillates…
In the statistical mechanics of quantum harmonic oscillators, the zero-point energy can either be included (Schr\"odinger oscillators) or omitted (Planck oscillators). For the usual results, the type of oscillator makes no difference but,…
We show that the PT symmetric Hamiltonians (and their generalizations defined in the text) may be all assigned the projected (so called Feshbach or effective) nonlinear Hamiltonians which are "locally" Hermitian. This implies that many (if…
We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The…
Non-Hermitian Hamiltonians, and particularly parity-time (PT) and anti-PT symmetric Hamiltonians, play an important role in many branches of physics, from quantum mechanics to optical systems and acoustics. Both the PT and anti-PT…
We consider PT-symmetric, discrete nonlocal nonlinear Schr\"{o}dinger equation on metric graphs. Soliton solutions are obtained for simplest graph topologies, such as star and tree graphs. Integrability of the problem is shown by proving…
An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. Numerical integrators using generating functions, Hamiltonian splitting,…
Stable discrete compactons in arrays of inter-connected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time PT symmetric models. The compactons result from the interference of the fields…
We study spectral and transport properties of one-dimensional tight-binding $\mathcal{PT}$-symmetric chains with alternating couplings. Based on the transfer matrix method, we have analytically developed the expressions for the transmission…
N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the…