arXiv subjects

Ivan Naumkin

Publications and source records attributed to Ivan Naumkin.

At least 19 recordsLinked to original sources

arXiv2026

On Matrix Valued Schr\"odinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, H\"older Regularity and Dispersive Estimates

This work establishes new results on spectral theory and time evolution for matrix-valued discrete Schr\"odinger operators on the space of square-summable matrix sequences. The matrix-valued formalism is employed to streamline notation, offering a more elegant alternative to the equivalent vector-valued framework with matrix potentials. Our main contributions are threefold: first, we derive an explicit Wronskian-based representation for the resolvent's integral kernel; second, we prove H\"older continuity for the resolvent's boundary values; and third, we establish dispersive estimates for the time evolution. Our approach begins with the construction of Jost solutions using Volterra equations and the transmutation operator, leading to proofs of their H\"older regularity and bounds in Wiener algebra norms. From these solutions, we obtain the explicit kernel representation. This explicit characterization - a distinctive feature of the one-dimensional setting - enables the direct computation of the resolvent's boundary values. We subsequently establish a limiting absorption principle and demonstrate the H\"older continuity of these boundary values, achieving an improvement over classical results obtained through abstract higher-dimensional methods. Finally, this detailed resolvent characterization is leveraged to prove the dispersive estimates for the time evolution of the system.

math.FA↗

arXiv2025

Reconstructing the Probability Measure of a Curie-Weiss Model Observing the Realisations of a Subset of Spins

We study the problem of reconstructing the probability measure of the Curie-Weiss model from a sample of the voting behaviour of a subset of the population. While originally used to study phase transitions in statistical mechanics, the Curie-Weiss or mean-field model has been applied to study phenomena, where many agents interact with each other. It is useful to measure the degree of social cohesion in social groups, which manifests in the way the members of the group influence each others' decisions. In practice, statisticians often only have access to survey data from a representative subset of a population. As such, it is useful to provide methods to estimate social cohesion from such data. The estimators we study have some positive properties, such as consistency, asymptotic normality, and large deviation principles. The main advantages are that they require only a sample of votes belonging to a (possibly very small) subset of the population and have a low computational cost. Due to the wide application of models such as Curie-Weiss, these estimators are potentially useful in disciplines such as political science, sociology, automated voting, and preference aggregation.

math.PR↗

arXiv2025

Approximation Techniques for the Reconstruction of the Probability Measure and the Coupling Parameters in a Curie-Weiss Model for Large Populations

The Curie-Weiss model, originally used to study phase transitions in statistical mechanics, has been adapted to model phenomena in social sciences where many agents interact with each other. Reconstructing the probability measure of a Curie-Weiss model via the maximum likelihood method runs into the problem of computing the partition function which scales exponentially with the population. We study the estimation of the coupling parameters of a multi-group Curie-Weiss model using large population asymptotic approximations for the relevant moments of the probability distribution in the case that there are no interactions between groups. As a result, we obtain an estimator which can be calculated at a low and constant computational cost for any size of the population. The estimator is consistent (under the added assumption that the population is large enough), asymptotically normal, and satisfies large deviation principles. The estimator is potentially useful in political science, sociology, automated voting, and in any application where the degree of social cohesion in a population has to be identified. The Curie-Weiss model's coupling parameters provide a natural measure of social cohesion. We discuss the problem of estimating the optimal weights in two-tier voting systems.

math.ST↗

arXiv2022

Non-flat conformal blow-up profiles for the 1D critical nonlinear Schrödinger equation

For the critical one-dimensional nonlinear Schrödinger equation, we construct blow-up solutions that concentrate a soliton at the origin at the conformal blow-up rate, with a non-flat blow-up profile. More precisely, we obtain a blow-up profile that equals $|x|+iκx^2$ near the origin, where $κ$ is a universal real constant. Such profile differs from the flat profiles obtained in the same context by Bourgain and Wang [Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997)].

math.AP↗

arXiv2022

Levinson theorem for discrete Schrödinger operators on the line with matrix potentials having a first moment

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

math-ph↗

arXiv2022

The Matrix Nonlinear Schrödinger Equation with a Potential

This paper is devoted to the study of the large-time asymptotics of the small solutions to the matrix nonlinear Schrödinger equation with a potential on the half-line and with general selfadjoint boundary condition, and on the line with a potential and a general point interaction, in the whole supercritical regime. We prove that the small solutions are scattering solutions that asymptotically in time, $t \to\pm\infty, $ behave as solutions to the associated linear matrix Schrödinger equation with the potential identically zero. The potential can be either generic or exceptional. Our approach is based on detailed results on the spectral and scattering theory for the associated linear matrix Schrödinger equation with a potential, and in a factorization technique that allows us to control the large-time behaviour of the solutions in appropriate norms.

math.AP↗

arXiv2020

$L^{p}-L^{p^{\prime}}$ estimates for matrix Schrödinger equations

This paper is devoted to the study of dispersive estimates for matrix Schrödinger equations on the half-line with general boundary condition, and on the line. We prove $L^{p}-L^{p^{\prime}}$ estimates on the half-line for slowly decaying selfadjoint matrix potentials that satisfy $\int_{0}^{\infty }\, (1+x) |V(x)|\, dx < \infty$ both in the generic and in the exceptional cases. We obtain our $L^{p}-L^{p^{\prime}}$ estimate on the line for a $n \times n$ system, under the condition that $\int_{-^{\infty}}^{\infty}\, (1+|x|)\, |V(x)|\, dx < \infty,$ from the $L^{p}-L^{p^{\prime}}$ estimate for a $2n\times2n$ system on the half-line. With our $L^{p}-L^{p^{\prime}}$ estimates we prove Strichartz estimates.

math-ph↗

arXiv2020

Asymptotic behavior for a dissipative nonlinear Schrödinger equation

We consider the Schrödinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +Δu=λ|u|^αu \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $λ\in {\mathbb C} $ with $\Imλ<0$. Assuming $\frac {2} {N+2}<α<\frac2N$, we give a precise description of the long-time behavior of the solutions (including decay rates in $L^2$ and $L^\infty $, and asymptotic profile), for a class of arbitrarily large initial data.

math.AP↗

arXiv2020

Sign-changing solutions of the nonlinear heat equation with persistent singularities

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = Δu + |u|^αu$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < α<α_0$, where $α_0=\frac {4} {N-4+2\sqrt{ N-1 } }\in (\frac {2} {N-2}, \frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $μ>0$ that can be arbitrarily large, we prove that for any $u_0 \in \mathrm{L} ^\infty _{\mathrm{loc}} ({\mathbb R}^N \setminus \{ 0 \}) $ which is bounded at infinity and equals $μ|x|^{- \frac {2} {α}}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $β|x|^{- \frac {2} {α}}$ at the origin in the sense that for $t>0$, $ |x|^{\frac {2} {α}} u(t,x) \to β$ as $ |x| \to 0$, where $β= \frac {2} {α} ( N -2 - \frac {2} {α} ) $. These solutions in general are neither stationary nor self-similar.

math.AP↗

arXiv2020

Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

We consider the nonlinear heat equation $u_t - Δu = |u|^αu$ on ${\mathbb R}^N$, where $α>0$ and $N\ge 1$. We prove that in the range $0 < α<\frac {4} {N-2}$, for every $μ>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= μ|x|^{-\frac {2} {α}}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.

math.AP↗

arXiv2020

Local smooth solutions of the nonlinear Klein-gordon equation

Given any $μ_1, μ_2\in {\mathbb C}$ and $α>0$, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $\partial_{ tt } u - Δu + μ_1 u = μ_2 |u|^αu$ on ${\mathbb R}^N$, $N\ge 1$, that do not vanish, i.e. $ |u (t,x) | >0 $ for all $x \in {\mathbb R}^N$ and all sufficiently small $t$. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from~[Commun. Contemp. Math. {\bf 19} (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.

math.AP↗

arXiv2018

On travelling waves of the non linear Schrödinger equation escaping a potential well

In this paper we consider the NLS equation with focusing nonlinearities in the presence of a potential. We investigate the compact soliton motions that correspond to a free soliton escaping the well created by the potential. We exhibit the dynamical system driving the exiting trajectory and construct associated nonlinear dynamics for untrapped motions. We show that the nature of the potential/soliton is fundamental, and two regimes may exist: one where the tail of the potential is fat and dictates the motion, one where the tail is weak and the soliton self interacts with the potential defects, hence leading to different motions.

math.AP↗

arXiv2018

Perturbations of self-similar solutions

We consider the nonlinear heat equation $u_t = Δu + |u|^αu$ with $α>0$, either on ${\mathbb R}^N $, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) α<4$, for every $μ\in {\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = μ|x-x_0|^{-\frac {2} {α}}$ in a neighborhood of some $x_0\in Ω$ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $μ|x-x_0|^{-\frac {2} {α}}$ on ${\mathbb R}^N $. Moreover, if $μ\ge μ_0$ for a certain $ μ_0( N, α)\ge 0$, and $u_0 I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.

math.AP↗

arXiv2018

Estimates in the modulation spaces for the Dirac equation with potential

In the present paper we obtain estimates in the modulation spaces for the solutions to the Dirac equation with quadratic and sub-quadratic potentials. We derive a representation for the Dirac operator that permits to solve approximately the perturbed Dirac equation and to obtain the desired estimates for the solution.

math.AP↗

arXiv2017

On small travelling waves to the mass critical fractional NLS

We consider the mass critical fractional (NLS). We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [Gerard P.; Lenzmann E.; Pocovnicu O.; Raphaël, P., A two soliton with transient turbulent regime for the one dimensional cubic half wave, submitted] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.

math.AP↗

arXiv2017

Nonlinear Schrödinger equations with exceptional potentials

We consider the cubic nonlinear Schrödinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of modified scattering for this model, that is, linear scattering modulated by a phase. Our approach is based on the spectral theorem for the perturbed linear Schrödinger operator and a factorization technique, that allows us to control the resonant nonlinear term. We make some parity assumptions in order to control the small-energy behavior of the scattering coefficients and of the wave functions.

math.AP↗

arXiv2017

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation $iu_t + Δu= λ|u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $λ\in {\mathbb C}$ and $\Im λ\le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty $, like $t^{- \frac {N} {2}}$ if $\Im λ=0$ and like $(t \log t)^{- \frac {N} {2}}$ if $\Im λ<0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty $. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.

math.AP↗

arXiv2016

Local existence, global existence, and scattering for the nonlinear Schrödinger equation

In this paper, we construct for every $α>0$ and $λ\in {\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schrödinger equation \begin{equation*} \begin{cases} iu_t + Δu + λ|u|^αu= 0 \\ u(0,x) = u_0 \end{cases} \end{equation*} on ${\mathbb R}^N $. Moreover, we construct for every $α>\frac {2} {N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty $.

math.AP↗