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We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
A hydrodynamic approach is used to calculate an asymptotics of the Emptiness Formation Probability - the probability of a formation of an empty space in the ground state of a quantum one-dimensional many body system. Quantum hydrodynamics…
In this thesis, I go through the well-known solutions to the one and two-particle systems trapped in a quantum harmonic oscillator and then continue to the three, four and many-body quantum systems. This is done by developing new analytical…
In this work simple and effective quantization procedure of classical dynamical systems is proposed and illustrated by a number of examples. The procedure is based entirely on differential equations which describe time evolution of systems.
We generalize the notion of partial dynamical symmetry (PDS) to a system of interacting bosons and fermions. In a PDS, selected states of the Hamiltonian are solvable and preserve the symmetry exactly, while other states are mixed. As a…
We study point and higher symmetries for the hydrodynamic-type systems with two independent variables $t$ and $x$ with and without explicit dependence of the equations on $t,x$. We consider those systems which possess an…
Probabilistic programming is related to a compositional approach to stochastic modeling by switching from discrete to continuous time dynamics. In continuous time, an operator-algebra semantics is available in which processes proceeding in…
The strict connection between Lie point-symmetries of a dynamical system and its constants of motion is discussed and emphasized, through old and new results. It is shown in particular how the knowledge of a symmetry of a dynamical system…
We examine the procedure to construct the variables of use for the momentum representation in quantum mechanics. The momentum variables must be chosen properly conjugate to the corresponding position space variables, such that valid…
We introduce a method of quantum tomography for a continuous variable system in position and momentum space. We consider a single two-level probe interacting with a quantum harmonic oscillator by means of a class of Hamiltonians, linear in…
We first consider the Hamiltonian formulation of $n=3$ systems in general and show that all dynamical systems in ${\mathbb R}^3$ are bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. We…
We present methods that can provide an exponential savings in the resources required to perform dynamic parameter estimation using quantum systems. The key idea is to merge classical compressive sensing techniques with quantum control…
A method of calculating a new class of symmetries is presented for partial differential equations. The method give a new dynamical solution for an isothermal and cylindrically symmetric hydrodynamics equations under self-gravity. The…
How do symmetries induce natural and useful quantum structures? This question is investigated in the context of models of three interacting particles in one-dimension. Such models display a wide spectrum of possibilities for dynamical…
A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed…
We propose a method to construct first integrals of a dynamical system, starting with a given set of independent infinitesimal symmetries. In the case of two infinitesimal symmetries, a rank two Poisson structure on the ambient space it is…
We discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system - or a parameter dependent family - via the Poincar\'e Normal Form approach. We discuss this at first formally, and…
Vortices are known to play a key role in the dynamics of the quantum trajectories defined within the framework of the de Broglie-Bohm formalism of quantum mechanics. It has been rigourously proved that the motion of a vortex in the…
In this paper, the Higgs-like approach is used to analyze the quantum dynamics of a harmonic oscillator constrained on a circle. We obtain the Hamiltonian of this system as a function of the Cartesian coordinate of the tangent line through…
The paper is devoted to further development of the new approach in equilibrium statistical mechanics the basis of which was worked out in a series of articles by the author. The approach proceeds on the use of a hierarchy of equations for…