It is obvious that we still have not any unified framework covering a zoo of interpretations of Quantum Mechanics,
as well as satisfactory understanding of main ingredients of a phenomena like entanglement. The starting point is
an idea to describe properly the key ingredient of the area, namely point/particle-like objects (physical quantum
points/particles or, at least, structureless but quantum objects) and to change point (wave) functions by sheaves
to the sheaf wave functions (Quantum Sheaves). In such an approach Quantum States are sections of the
coherent sheaves or contravariant functors from the kinematical category describing space-time to other one,
Quantum Dynamical Category, properly describing the complex dynamics of Quantum Patterns. The objects
of this category are some filtrations on the functional realization of Hilbert space of Quantum States. In this
Part 2, the sequel of Part 1, we present a family of methods which can describe important details of complex
behaviour in quantum ensembles: the creation of nontrivial patterns, localized, chaotic, entangled or decoherent,
from the fundamental basic localized (nonlinear) eigenmodes (in contrast with orthodox gaussian-like) in various
collective models arising from the quantum hierarchies described by Wigner-like equations.
We consider some generalization of the theory of quantum states, which is based on the analysis of long standing
problems and unsatisfactory situation with the possible interpretations of quantum mechanics. We demonstrate
that the consideration of quantum states as sheaves can provide, in principle, more deep understanding of some
phenomena. The key ingredients of the proposed construction are the families of sections of sheaves with values
in the category of the functional realizations of infinite-dimensional Hilbert spaces with special (multiscale)
filtration. Three different symmetries, kinematical (on space-time), hidden/dynamical (on sections of sheaves),
unified (on filtration of the full scale of spaces) are generic objects generating the full zoo of quantum phenomena.
In this second part we present a set of methods, analytical and
numerical, which can describe behaviour in (non) equilibrium ensembles, both classical and quantum, especially in the complex systems, where the standard approaches cannot be applied. The key points demonstrating advantages of this approach are: (i) effects of localization of possible quantum states; (ii) effects of non-perturbative multiscales which cannot be calculated by means of perturbation approaches; (iii) effects of formation of complex/collective quantum patterns from localized modes and classification and possible control of the full zoo of quantum states, including (meta) stable localized patterns (waveletons). We demonstrate the appearance of nontrivial localized (meta) stable states/patterns in a number of collective models covered by the
(quantum)/(master) hierarchy of Wigner-von Neumann-Moyal-Lindblad equations, which are the result of "wignerization" procedure (Weyl-Wigner-Moyal quantization) of classical BBGKY kinetic hierarchy, and present the explicit constructions for exact analytical/numerical computations. Our fast and efficient approach is based on variational and multiresolution representations in the bases of polynomial tensor algebras of generalized localized states (fast convergent variational-wavelet representation). We construct the representations for hierarchy/algebra of observables(symbols)/distribution functions via the complete multiscale decompositions, which allow to consider the polynomial and rational type of nonlinearities. The solutions are represented via the exact decomposition in nonlinear high-localized eigenmodes, which correspond to the full multiresolution expansion in all underlying hidden time/space or phase space scales. In contrast with different approaches we do not use perturbation technique or linearization procedures. Numerical modeling shows the creation of different internal structures from localized modes, which are related to the localized (meta) stable patterns (waveletons), entangled ensembles (with subsequent decoherence) and/or chaotic-like type of behaviour.
In these two related parts we present a set of methods, analytical and numerical, which can illuminate the behaviour of quantum system, especially in the complex systems, e.g., where the standard "coherent-states" approach cannot be applied. The key points demonstrating advantages of this approach are: (i) effects of localization of possible quantum states, more proper than "gaussian-like states";
(ii) effects of non-perturbative multiscales which cannot be calculated by means of perturbation approaches; (iii) effects of formation of complex quantum patterns from localized modes or classification and possible control of the full zoo of quantum states, including (meta) stable localized patterns (waveletons). In this first part we consider the applications of numerical-analytical technique based on local nonlinear harmonic analysis to quantum/quasiclassical description of nonlinear (polynomial/rational) dynamical problems which appear in many areas of physics. We'll consider calculations of Wigner functions as the solution of Wigner-Moyal-von Neumann equation(s) corresponding to polynomial Hamiltonians. Modeling demonstrates the appearance of (meta) stable patterns generated by high-localized (coherent) structures or entangled/chaotic behaviour. We can control the type of behaviour on the level of reduced algebraical variational system. At the end we presented the qualitative definition of the Quantum Objects in comparison with their Classical Counterparts, which natural domain of definition is the category of multiscale/multiresolution decompositions according to the action of internal/hidden symmetry of the proper realization of scales of functional spaces (the multiscale decompositions of the scales of Hilbert spaces of states). It gives rational natural explanation of such pure quantum effects as "self-interaction" (self-interference) and instantaneous quantum interaction (transmission of information).