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Dmitry Treschev

Publications and source records attributed to Dmitry Treschev.

At least 19 recordsLinked to original sources
arXiv2026

Isonormal potentials in one degree of freedom

We consider the Hamiltonian system with one degree of freedom $$\dot x = y, \quad \dot y = -\partial V(x) / \partial x, \qquad x,y\in{\mathbb R}$$ with the smooth potential $V:{\mathbb R}\to{\mathbb R}$. We assume that the origin is an elliptic singular point: $V'(0)=0$ and $V''(0)>0$. Two potentials $V_0$ and $V_1$ are referred to be isonormal if in a neighborhood of the origin there exists a canonical change of coordinates which transforms the Hamiltonian function $y^2/2 + V_0$ to $y^2/2 + V_1$. In particular, any potential isonormal to $x^2/2$ is isochronous. We obtain a criterium of isonormality for analytic potentials.

math.DS↗
arXiv2026

Integrable perturbations of polynomial Hamiltonian systems

We consider a Hamiltonian system on the symplectic space $({\mathbb{R}}^{2n}, dy\wedge dx)$ with a real-analytic Hamiltonian $H : {\mathbb{R}}^{2n}\to {\mathbb{R}}$. We assume that the system has a non-degenerate equilibrium position at the origin. Under some nonresonance assumptions we prove the following. For any positive integer $M$ there exists a real-analytic function $F:{\mathbb{R}}^{2n}\to{\mathbb{R}}$ such that (1) $F = O\big( (|x|+|y|)^{M+1} \big)$ at the origin, (2) the system with Hamiltonian $H+F$ is completely integrable in ${\mathbb{R}}^{2n}$.

math.DS↗
arXiv2026

Combinatorics of Hamiltonian Normal Forms

We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point purely in terms of the original Hamiltonian, avoiding the normalization procedure. In the case of one degree of freedom we compute the normal form as an explicit nonlinear functional, applied to the original Hamiltonian. We present analogous results in arbitrary dimension. The corresponding formulas are more complicated but still explicit.

math.DS↗
arXiv2025

On the problem of stability of viscous shocks

We consider the problem of spectral stability of traveling wave solutions $u=\gamma(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $\gamma$ of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields $F$ for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.

math.AP↗
arXiv2025

Another billiard problem

Let $(M,g)$ be a Riemannian manifold, $\Omega\subset M$ a domain with boundary $\Gamma$, and $\phi$ a smooth function such that $\phi|_\Omega > 0$, $\ph|_\Gamma = 0$, and $\nabla\phi|_\Gamma\ne 0$. We study the geodesic flow of the metric $G=g/\phi$. The $G$-distance from any point of $\Omega$ to $\Gamma$ is finite, hence the geodesic flow is incomplete. Regularization of the flow in a neighborhood of $\Gamma$ establishes a natural reflection law from $\Gamma$. This leads to a certain billiard problem in $\Omega$.

math.DS↗
arXiv2024

On quantum Floquet theorem

We consider the Schr\"odinger equation $ih\partial_t\psi = H\psi$, $\psi=\psi(\cdot,t)\in L^2({\mathbb T})$. The operator $H = -\partial^2_x + V(x,t)$ includes smooth potential $V$, which is assumed to be time $T$-periodic. Let $W=W(t)$ be the fundamental solution of this linear ODE system on $L^2({\mathbb T})$. Then according to terminology from Lyapunov-Floquet theory, ${\cal M}=W(T)$ is the monodromy operator. We prove that ${\cal M}$ is unitarily conjugated to $\exp\big(-\frac{T}{ih} \partial^2_x\big) + {\cal C}$, where ${\cal C}$ is a compact operator with an arbitrarily small norm.

math.DS↗
arXiv2024

Normalization flow in the presence of a resonance

Following arXiv:2303.02992, we develop an approach to the Hamiltonian theory of normal forms based on continuous averaging. We concentrate on the case of normal forms near an elliptic singular point, but unlike arXiv:2303.02992 we do not assume that frequences of the linearized system are nonresonant. We study analytic properties of the normalization procedure. In particular we show that in the case of a codimension one resonance an analytic Hamiltonian function may be reduced to a normal form up to an exponentially small reminder with explicit estimates of the reminder and the analyticity domain.

math.DS↗
arXiv2023

Normalization flow

We propose a new approach to the theory of normal forms for Hamiltonian systems near a non-resonant elliptic singular point. We consider the space of all Hamiltonian functions with such an equilibrium position at the origin and construct a differential equation in this space. Solutions of this equation move Hamiltonian functions towards their normal forms. Shifts along the flow of this equation correspond to canonical coordinate changes. So, we have a continuous normalization procedure. The formal aspect of the theory presents no difficulties. The analytic aspect and the problems of convergence of series are as usual non-trivial.

math.DS↗
arXiv2021

$μ$-norm and regularity

In \cite{Tre_PSI20} we introduce the concept of a $μ$-norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from \cite{Tre_PSI20} and present further results on the $μ$-norm. More precisely, we specify three classes of unitary operators for which the $μ$-norm generates a bistochastic operator. We plan to use the latter in the construction of quantum entropy.

math.DS↗
arXiv2021

$μ$-norm of an operator

Let $({\cal X},μ)$ be a measure space. For any measurable set $Y\subset{\cal X}$ let $1_Y : {\cal X}\to{\mathbb R}$ be the indicator of $Y$ and let $π_Y$ be the orthogonal projector $L^2({\cal X})\ni f\mapstoπ_Y f = 1_Y f$. For any bounded operator $W$ on $L^2({\cal X},μ)$ we define its $μ$-norm $\|W\|_μ= \inf_χ\sqrt{\sum μ(Y_j) \|Wπ_Y\|^2}$, where the infinum is taken over all measurable partitions $χ= \{Y_1,\ldots,Y_J\}$ of ${\cal X}$. We present some properties of the $μ$-norm and some computations. Our main motivation is the problem of the construction of a quantum entropy.

math.DS↗
arXiv2021

Isochronicity in 1 DOF

Our main result is the complete set of explicit conditions necessary and sufficient for isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented in terms of Taylor coefficients of the Hamiltonian function.

math.DS↗
arXiv2018

Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a non-autonomous near-integrable Hamiltonian system with $n+1/2$ degrees of freedom, $n\ge 2$. Let the Hamilton function $H$ of depend on the parameter $\varepsilon$, for $\varepsilon=0$ the system is integrable and has a homoclinic asymptotic manifold $Γ$. Our main result is that for small generic perturbation in an $\varepsilon$-neighborhood of $Γ$ there exist trajectories the projections of which on the space of actions cross the resonance. By ``generic perturbations'' we mean an open dense set in the space of $C^r$-smooth functions $\frac{d}{d\varepsilon}\big|_{\varepsilon=0} H$, $r=r_0,r_0+1,\ldots,\infty,ω$. Combination of this result with results of \cite{DT} answers the main questions on the Arnold diffusion in a priori unstable case: the diffusion takes place for generic perturbation, diffusion trajectories can go along any smooth curve in the action space with average velocity of order $\varepsilon/|\log \varepsilon|$.

math.DS↗
arXiv2016

A locally integrable multi-dimensional billiard system

We consider a multi-dimensional billiard system in an (n+1)-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $γ$ of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $γ$) conjugated to the dynamics of a linear map? Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions $\pm f$, where $f$ is even and defined in a neighborhood of the origin on the "horizontal" coordinate hyperplane. We prove that $f$ exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.

math.DS↗
arXiv2016

Anti-integrable limit

Anti-integrable limit is one of convenient and relatively simple methods for construction of chaotic hyperbolic invariant sets in Lagrangian, Hamiltonian and other dynamical systems. We discuss the most natural context of the method -- discrete Lagrangian systems. Then we present examples and applications.

math.DS↗
arXiv2013

Lagrangian tori near resonances of near-integrable Hamiltonian systems

In this paper we study families of Lagrangian tori that appear in a neighborhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the "integrable" limit $\varepsilon\to 0$. Dynamics on these tori is oscillatory in the direction of the resonance phases and rotating with respect to the other (non-resonant) phases. We also show that, if multiplicity of the resonance equals one, generically these tori occupy a set of large relative measure in the resonant domains in the sense that the relative measure of the remaining "chaotic" set is of order $\sqrt\varepsilon$. Therefore for small $\varepsilon > 0$ a random initial condition in a $\sqrt\varepsilon$-neighborhood of a single resonance occurs inside this set (and therefore generates a quasi-periodic motion) with a probability much larger than in the "chaotic" set. We present results of numerical simulations and discuss the form of projection of such tori to the action space.

math.DS↗
arXiv2013

On the support of a body by a surface with random roughness

Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points provided the roughness is fine grained. We also solve an analogous problem where a circle is put on a rough plane. Some applications in static are given.

math.PR↗
arXiv2012

On Collisions in Nonholonomic systems

We consider nonholonomic systems with collisions and propose a concept of weak solutions to Lagrange-d'Alembert equations. In the light of this concept we describe dynamics of the collisions. Several applications have been investigated. Particularly the collision of rotating ball and the rough floor has been considered.

math-ph↗