arXiv subjects

Vedran Sohinger

Publications and source records attributed to Vedran Sohinger.

At least 19 recordsLinked to original sources

arXiv2026

Bounds on the growth of high Sobolev norms of solutions to the fractional nonlinear Schr\"{o}dinger equation on $\R^2$ and $\R^3$

We prove polynomial bounds on the growth of high Sobolev norms of solutions to the defocusing fractional nonlinear Schr\"odinger equation with cubic and Hartree-type nonlinearities in two and three dimensions. Our result is based on the upside-down I-method and the method of higher modified energies. The analysis of the fractional case in higher dimensions is possible due to a higher-dimensional analogue of a resonance inequality, which allows us to control the nonresonant frequency contributions.

math.AP↗

arXiv2026

The large-mass limit of interacting quantum gases in the continuum

We study the large-mass limit of interacting quantum (Bose or Fermi) gases in thermal equilibrium. We show that in the suitably-defined large-mass limit, the system gives rise to a gas of classical interacting particles. The corresponding question for bosons on a lattice was previously addressed by Fr\"{o}hlich, Knowles, Schlein, and the third author. In this work, we study the continuum regime which requires us to suitably tune the chemical potential. The starting point of our analysis is the Ginibre loop ensemble, which allows one to describe a system of interacting quantum gases in thermal equilibrium in terms of an ensemble of interacting Brownian paths. In a finite volume, our analysis is performed for stable and H\"{o}lder continuous interaction potentials and we are able to obtain explicit rates of convergence. When the interaction potential is nonnegative and satisfies suitable integrability conditions, we study the associated infinite-volume problem by means of cluster expansions.

math-ph↗

arXiv2024

Gibbs measures as local equilibrium KMS states for focusing nonlinear Schr\"odinger equations

In this paper, we are concerned with the study of statistical equilibria for focusing nonlinear Schr\"odinger and Hartree equations on the d-dimensional torus when d=1,2,3. Due to the focusing nature of the nonlinearity in these PDEs, Gibbs measures have to be appropriately localized. First, we show that these local Gibbs measures are stationary solutions for the Liouville probability density equation and that they satisfy a local equilibrium Kubo-Martin-Schwinger (KMS) condition. Secondly, under some natural assumptions, we characterize all possible local KMS equilibrium states for these PDEs as local Gibbs measures. Our methods are based on Malliavin calculus in Gross-Stroock Sobolev spaces and on a suitable Gaussian integration by parts formula. To handle the technical problems due to localization, we rely on the works of Aida and Kusuoka on irreducibility of Dirichlet forms over infinite-dimensional domains. This leads us to the study of sublevel sets of the renormalized mass and their connectedness properties. In this paper, we also revisit Bourgain's proof of the normalizability of the local Gibbs measure for the focusing Hartree equation on the d-dimensional torus with d=2,3 by using concentration inequalities.

math.AP↗

arXiv2023

A microscopic derivation of Gibbs measures for the 1D focusing quintic nonlinear Schr\"{o}dinger equation

In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{T}$ from many-body quantum Gibbs states. On the quantum many-body level, the quintic nonlinearity corresponds to a three-body interaction. This is a continuation of our previous work. In the aforementioned work, we studied the cubic problem, which corresponds to a two-body interaction on the quantum many-body level. In our setup, we truncate the mass of the classical free field in the classical setting and the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion previously developed in the work of Fr\"{o}hlich, Knowles, Schlein, and the second author. We prove results both in the time-independent and time-dependent setting. This is the first such known result in the three-body regime. Furthermore, this gives the first microscopic derivation of time-dependent correlation functions for Gibbs measures corresponding to the quintic NLS, as studied in the work of Bourgain.

math-ph↗

arXiv2023

Almost sure existence of global solutions for general initial value problems

This article is concerned with the almost sure existence of global solutions for initial value problems of the form $\dot{\gamma}(t)= v(t,\gamma(t))$ on separable dual Banach spaces. We prove a general result stating that whenever there exists $(\mu_t)_{t\in \mathbb{R}}$ a family of probability measures satisfying a related statistical Liouville equation, there exist global solutions to the initial value problem for $\mu_0$-almost all initial data, possibly without uniqueness. The main assumption is a mild integrability condition of the vector field $v$ with respect to $(\mu_t)_{t\in \mathbb{R}}$. As a notable application, we obtain from the above principle that Gibbs and Gaussian measures yield low regularity global solutions for several nonlinear dispersive PDEs as well as fluid mechanics equations including the Hartree, Klein-Gordon, NLS, Euler and modified surface quasi-geostrophic equations. In this regard, our result generalizes Bourgain's method as well as Albeverio & Cruzeiro's method of constructing low regularity global solutions, without the need for local well-posedness analysis.

math.AP↗

arXiv2022

A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schr\"{o}dinger equation

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schr\"odinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in previous work of Fr\"ohlich, Knowles, Schlein, and the second author. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, integrable, and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.

math-ph↗

arXiv2022

The Euclidean $\phi^4_2$ theory as a limit of an interacting Bose gas

We prove that the complex Euclidean field theory with local quartic self-interaction in two dimensions arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The field theory is supported on distributions of negative regularity, which requires a renormalization by divergent mass and energy counterterms. We obtain convergence of the relative partition function and uniform convergence of the renormalized reduced density matrices. The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced in [32] using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions. As a byproduct of our proof, in two and three dimensions we also extend the results on the mean-field limit from [32,56] to unbounded interaction potentials satisfying the optimal integrability conditions proposed by Bourgain [13].

math-ph↗

arXiv2021

Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs

The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrodinger, Hartree, and wave (Klein-Gordon) equations.

math.PR↗

arXiv2020

Interacting loop ensembles and Bose gases

We study interacting Bose gases in thermal equilibrium on a lattice. We establish convergence of the grand canonical Gibbs states of such gases to their mean-field (classical field) and large-mass (classical particle) limits. The former is a classical field theory for a complex scalar field with quartic self-interaction. The latter is a classical theory of point particles with two-body interactions. Our analysis is based on representations in terms of ensembles of interacting random loops, the Ginibre loop ensemble for Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, our results also hold in infinite volume.

math-ph↗

arXiv2020

A path-integral analysis of interacting Bose gases and loop gases

We review some recent results on interacting Bose gases in thermal equilibrium. In particular, we study the convergence of the grand-canonical equilibrium states of such gases to their mean-field limits, which are given by the Gibbs measures of classical field theories with quartic Hartree-type self-interaction, and to the Gibbs states of classical gases of point particles. We discuss various open problems and conjectures concerning, e.g., Bose-Einstein condensation, polymers and $\vert \boldsymbolϕ \vert^{4}$-theory.

math-ph↗

arXiv2020

The mean-field limit of quantum Bose gases at positive temperature

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schr\"odinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leq 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.

math-ph↗

arXiv2019

A microscopic derivation of Gibbs measures for nonlinear Schrödinger equations with unbounded interaction potentials

We study the derivation of the Gibbs measure for the nonlinear Schrödinger equation (NLS) from many-body quantum thermal states in the high-temperature limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\mathbb{T}^d$ for $d=1,2,3$. This extends the author's earlier joint work with Fröhlich, Knowles, and Schlein, where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin, Nam, and Rougerie. Our proof is based on a perturbative expansion in the interaction. When $d=1$, the thermal state is the grand canonical ensemble. As in the author's earlier joint work with Fröhlich, Knowles, and Schlein, when $d=2,3$, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of $p$ and obtain the full range of defocusing interaction potentials which were studied in the classical setting when $d=2,3$ in the work of Bourgain.

math.AP↗

arXiv2019

Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d \leq 3$

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions $d =1,2,3$. The many-body quantum thermal states that we consider are the grand canonical ensemble for $d = 1$ and an appropriate modification of the grand canonical ensemble for $d =2,3$. In dimensions $d =2,3$, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

math-ph↗

arXiv2018

Unconditional Uniqueness Results for the Nonlinear Schrödinger Equation

We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation. We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross-Pitaevskii hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the Gross-Pitaevskii hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier-Lebesgue spaces implies unconditional uniqueness.

math.AP↗

arXiv2017

A microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schrödinger equation

We give a microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is our previous work on the time-independent problem and work of the second author on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by the means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in $X^{s,b}$ spaces.

math-ph↗

arXiv2014

The Gross-Pitaevskii hierarchy on general rectangular tori

In this work, we study the Gross-Pitaevskii hierarchy on general --rational and irrational-- rectangular tori of dimension two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in previous work of Kirkpatrick, Schlein, and Staffilani.

math.AP↗

arXiv2014

Randomization and the Gross-Pitaevskii hierarchy

We study the Gross-Pitaevskii hierarchy on the spatial domain $\mathbb{T}^3$. By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $α>\frac{3}{4}$. It was shown in our previous joint work with Gressman that the range $α>1$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon, in which the spacetime estimate is known to hold whenever $α\geq 1$. The goal of our paper is to extend the range of $α$ in this class of estimates in a \emph{probabilistic sense}. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

math.AP↗

arXiv2014

Local existence of solutions to Randomized Gross-Pitaevskii hierarchies

In this paper, we study the local-in-time existence of solutions to randomized forms of the Gross-Pitaevskii hierarchy on periodic domains. In particular, we study the independently randomized Gross-Pitaevskii hierarchy and the dependently randomized Gross-Pitaevskii hierarchy, which were first introduced in the author's joint work with Staffilani \cite{SoSt}. For these hierarchies, we construct local-in-time low-regularity solutions in spaces which contain a random component. The constructed density matrices will solve the full randomized hierarchies, thus extending the results from \cite{SoSt}, where solutions solving arbitrarily long subhierarchies were given. Our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of T. Chen and Pavlović \cite{CP4}. The presence of randomization in the problem adds additional difficulties, most notably to estimating the Duhamel expansions that are crucial in the truncation argument. These difficulties are overcome by a detailed analysis of the Duhamel expansions. In the independently randomized case, we need to keep track of which randomization parameters appear in the Duhamel terms, whereas in the dependently randomized case, we express the Duhamel terms directly in terms of the initial data. In both cases, we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data.

math.AP↗