Abstract:
Neutrino oscillations in matter can exhibit a specific resonance enhancement -- parametric resonance, which is different from the MSW resonance. Oscillations of atmospheric and solar neutrinos inside the earth can undergo parametric enhancement when neutrino trajectories cross the core of the earth. In this paper we review the parametric resonance of neutrino oscillations in matter. In particular, physical interpretation of the effect and the prospects of its experimental observation in oscillations of solar and atmospheric neutrinos in the earth are discussed.

Abstract:
In this paper, we study a photon, a massive charged particle, and a massive neutrino passing through matter. The hypothetical left-right symmetric weak interaction, which is used in Wolfenstein's equation can generate the resonance enhancement of neutrino oscillations in matter, which disappears when neutrinos go out into vacuum from matter (the Sun). It is shown that since standard weak interactions cannot generate masses, the laws of conservation of the energy and the momentum of neutrino in matter will be fulfilled only if the energy $W$ of polarization of matter by the neutrino or the corresponding term in Wolfenstein's equation, is zero. This result implies that neutrinos cannot generate permanent polarization of matter. This leads to the conclusion: resonance enhancement of neutrino oscillations in matter does not exist. It is also shown that in standard weak interactions the Cherenkov radiation cannot exist.

Abstract:
We study the parametric resonance of the neutrino oscillation through the matter whose density varies spatially. The Fourier analysis of the matter effect enables us to clarify the parametric resonance condition, which is summarized in a frequency matching between the neutrino oscillation and the spatial variation of the matter density. As a result, the n-th Fourier mode of a matter density profile modifies the energy spectrum of the nu_mu -> nu_e appearance probability at around the n-th dip.

Abstract:
Accretion disks arising from neutron star- neutron star mergers or black hole- neutron star mergers produce large numbers of neutrinos and antineutrinos. In contrast to other astrophysical scenarios, like supernovae, in mergers the antineutrinos outnumber the neutrinos. This antineutrino dominance gives neutrinos from merger disks the opportunity to exhibit new oscillation physics, specifically a matter-neutrino resonance. We explore this resonance, finding that consequences can be a large transition of $\nu_e$ to other flavors, while the $\bar{\nu}_e$s return to their initial state. We present numerical calculations of neutrinos from merger disks and compare with a single energy model. We explain both the basic features and the conditions for a transition.

Abstract:
The mechanism of resonance enhancement of neutrino oscillations in matter and some critical remarks to this mechanism are considered. Using of this resonance mechanism is very important to examine the model of electroweak interactions since the processes induced by this mechanism grow multiply. In contrast to the electromagnetic and strong interactions in weak interactions, $P$-parity is violated therefore a problem of mass generations in the weak interactions is considered (the interaction must be left-right symmetric for mass generations). It is concluded that a possibility of mass generation in the framework of the weak interactions is not proved. The present experimental status of this resonance mechanism is considered and it is done conclusion that this effect has no clear experimental confirmation. For this purpose it is necessary to fulfil precision experiments with solar neutrinos and the neutrinos passed through the Earth matter.

Abstract:
Neutrino oscillations in matter can exhibit a specific resonance enhancement -- parametric resonance, which is different from the MSW resonance. Recently it has been shown that the oscillations of atmospheric and solar neutrinos inside the earth can undergo parametric enhancement when neutrino trajectories cross the core of the earth. In this paper we continue the study of the parametric resonance of neutrino oscillations in the earth. The following issues are discussed: stability of the resonance with respect to the variations of the zenith angle of the neutrino source; higher-order resonances; prospects of the experimental observation of the parametric resonance of neutrino oscillations. We also comment on a recent controversy regarding the physical nature of the resonance enhancement of the oscillations of the core crossing neutrinos in the earth.

Abstract:
Within the Lorentz invariant formalizm for description of neutrino evolution in electromagnetic fields and matter we consider neutrino spin oscillations in the circular polarized electromagnetic wave, the amplitude of which is a modulated function of time. It is shown for the first time that the parametric resonance of neutrino oscillations can occur in such a system.

Abstract:
We derive analytic expressions for three flavor neutrino oscillations in the presence of matter in the plane wave approximation using the Cayley-Hamilton formalism. Especially, we calculate the time evolution operator in both flavor and mass bases. Furthermore, we find the transition probabilities, matter mass squared differences, and matter mixing angles all expressed in terms of the vacuum mass squared differences, the vacuum mixing angles, and the matter density. The conditions for resonance in the presence of matter are also studied in some examples.

Abstract:
We study the system of massive and mixed neutrinos interacting with background matter moving with an acceleration. We start with the derivation of the Dirac equation for a single neutrino in the noninertial frame where matter is at rest. A particular case of matter rotating with a constant angular velocity is considered. The Dirac equation is solved and the neutrino energy levels are found for ultrarelativistic particles propagating in rotating matter. Then we generalize our results to include several neutrino generations and consider mixing between them. Using the relativistic quantum mechanics approach we derive the effective Schr\"{o}dinger equation for the description of neutrino flavor oscillations in rotating matter. We obtain the resonance condition for neutrino oscillations and examine how it can be affected by the matter rotation. We also compare our results with the findings of other authors who studied analogous problem previously.

Abstract:
A simple closed-form analytic expression for the probability of two-flavour neutrino oscillations in a matter with an arbitrary density profile is derived. Our formula is based on a perturbative expansion and allows an easy calculation of higher order corrections. The expansion parameter is small when the density changes relatively slowly along the neutrino path and/or neutrino energy is not very close to the Mikheyev-Smirnov-Wolfenstein (MSW) resonance energy. Our approximation is not equivalent to the adiabatic approximation and actually goes beyond it. We demonstrate the validity of our results using a few model density profiles, including the PREM density profile of the Earth. It is shown that by combining the results obtained from the expansions valid below and above the MSW resonance one can obtain a very good description of neutrino oscillations in matter in the entire energy range, including the resonance region.