Oscillatory and non-oscillatory criteria for linear four-dimensional Hamiltonian systems

summary:The Riccati equation method is used to study the oscillatory and non-oscillatory behavior of solutions of linear four-dimensional Hamiltonian systems. One oscillatory and three non-oscillatory criteria are proved. Examples of the obtained results are compared with some well known ones

Correlation-induced localization

A new paradigm of Anderson localization caused by correlations in the long-range hopping along with uncorrelated on-site disorder is considered which requires a more precise formulation of the basic localization-delocalization principles. A new class of random Hamiltonians with translation-invariant hopping integrals is suggested and the localization properties of such models are established both in the coordinate and in the momentum spaces alongside with the corresponding level statistics. Duality of translation-invariant models in the momentum and coordinate space is uncovered and exploited to find a full localization-delocalization phase diagram for such models. The crucial role of the spectral properties of hopping matrix is established and a new matrix inversion trick is suggested to generate a one-parameter family of equivalent localization/delocalization problems. Optimization over the free parameter in such a transformation together with the localization/delocalization principles allows to establish exact bounds for the localized and ergodic states in long-range hopping models. When applied to the random matrix models with deterministic power-law hopping this transformation allows to confirm localization of states at all values of the exponent in power-law hopping and to prove analytically the symmetry of the exponent in the power-law localized wave functions.Comment: 14 pages, 8 figures + 5 pages, 2 figures in appendice

Collective coherence in planar semiconductor microcavities

Semiconductor microcavities, in which strong coupling of excitons to confined photon modes leads to the formation of exciton-polariton modes, have increasingly become a focus for the study of spontaneous coherence, lasing, and condensation in solid state systems. This review discusses the significant experimental progress to date, the phenomena associated with coherence which have been observed, and also discusses in some detail the different theoretical models that have been used to study such systems. We consider both the case of non-resonant pumping, in which coherence may spontaneously arise, and the related topics of resonant pumping, and the optical parametric oscillator.Comment: 46 pages, 12 figure

The role of fluctuations in quantum and classical time crystals

Discrete time crystals (DTCs) are a many-body state of matter whose dynamics are slower than the forces acting on it. The same is true for classical systems with period-doubling bifurcations. Hence, the question naturally arises what differentiates classical from quantum DTCs. Here, we analyze a variant of the Bose-Hubbard model, which describes a plethora of physical phenomena and has both a classical and a quantum time-crystalline limit. We study the role of fluctuations on the stability of the system and find no distinction between quantum and classical DTCs. This allows us to probe the fluctuations in an experiment using two strongly coupled parametric resonators subject to classical noise.Comment: 11 pages, 5 figure