arXiv subjects

Valery Romanovski

Publications and source records attributed to Valery Romanovski.

6 recordsLinked to original sources

arXiv2024

Integrability of polynomial vector fields and a dual problem

We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the intrinsic connection between these operators and local analytic first integrals. To illustrate the duality the case of quadratic vector fields is considered in detail.

nlin.SI↗

arXiv2024

Integrability and Linearizability of a Family of Three-Dimensional Polynomial Systems

We investigate the local integrability and linearizability of a family of three-dimensional polynomial systems with the matrix of the linear approximation having the eigenvalues $1, ζ, ζ^2 $, where $ζ$ is a primitive cubic root of unity. We establish a criterion for the convergence of the Poincaré--Dulac normal form of the systems and examine the relationship between the normal form and integrability. Additionally, we introduce an efficient algorithm to determine the necessary conditions for the integrability of the systems. This algorithm is then applied to a quadratic subfamily of the systems to analyze its integrability and linearizability. Our findings offer insights into the integrability properties of three-dimensional polynomial systems.

math.DS↗

arXiv2002

Semiclassical analysis of Wigner functions

In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of \hbar can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of \hbar.

nlin.CD↗

arXiv2001

The Center Variety of Polynomial Differential Systems

We investigate the symmetry component of the center variety of polynomial differential systems, corresponding to systems with an axis of symmetry in the real plane. We give a general algorithm to find this irreducible subvariety and compute its dimension. We show that our methods provide a simple way to compute the radical of the ideal generated by the focus quantities and, therefore, to estimate the cyclicity of a center in the case when the ideal is radical. In particular, we use our methods to get a simple proof of the famous Bautin theorem on the cyclicity of the quadratic system.

math.DS↗

arXiv2000

On WKB Series for the Radial Kepler Problem

We obtain the rigorous WKB expansion to all orders for the radial Kepler problem, using the residue calculus in evaluating the WKB quantization condition in terms of a complex contour integral in the complexified coordinate plane. The procedure yields the exact energy spectrum of this Schrödinger eigenvalue problem and thus resolves the controversies around the so-called "Langer correction". The problem is nontrivial also because there are only a few systems for which all orders of the WKB series can be calculated, yielding a convergent series whose sum is equal to the exact result, and thus sheds new light to similar and more difficult problems.

nlin.CD↗

arXiv2000

On the semiclassical expansion for 1-dim $x^N$ potentials

In the present paper we study the structure of the WKB series for the polynomial potential $V(x)=x^N$ ($N$ even). In particular, we obtain relatively simple recurrence formula of the coefficients $\s'_k$ of the semiclassical approximation and of the WKB terms for the energy eigenvalues.

nlin.CD↗