arXiv subjects

Andrew Rout

Publications and source records attributed to Andrew Rout.

7 recordsLinked to original sources

arXiv2026

A perturbative microscopic derivation of the focusing $\Phi^6_1$ measure with rough cut-off

We give a derivation of the Gibbs measure for the focusing nonlinear Schr\"odinger equation (NLS) on the circle with rough cut-off. This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs. Our proof is based on the perturbative expansion developed by Fr\"ohlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in L\"u, Nam, and Zhu (2026). To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for the cut-off. In particular, we give a derivation of the Gibbs measure for the focusing quintic NLS with the optimal cut-off from Oh, Sosoe, and Tolomeo (2022).

math-ph↗

arXiv2026

Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle

In this paper, we introduce a renormalisation procedure for the density associated with the system of nonlinear Schr\"odinger equations (NLSS) on a circle. We show that this renormalised density satisfies better orthonormal Strichartz estimates than the non-renormalised density, which was considered in Nakamura (2020). As an application, we determine the critical Schatten exponent below which the cubic renormalised NLSS on the circle is globally well-posed and above which it is ill-posed. Finally, we show that the improvement for orthonormal Strichartz estimates satisfied by the renormalised density on $\mathbb{T}^d$ for $d \ge 2$ is minimal.

math.AP↗

arXiv2025

Well-posedness of the periodic nonlinear Schr\"odinger equation with concentrated nonlinearity

We study the solution theory of the nonlinear Schr\"odinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. We also prove local well-posedness of the concentrated nonlinear Schr\"odinger equation below the energy space but above the endpoint of Sobolev embedding theorem. Our methods are based on compactness results and exploiting the Volterra integral equation structure of the problem. To our knowledge, this is the first rigorous solution theory for the concentrated nonlinear Schr\"odinger equation on the circle.

math.AP↗

arXiv2025

The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$

In this article, we consider Hartree equations generalised to $2p+1$ order nonlinearities. These equations arise in the study of the mean-field limits of Bose gases with $p$-body interactions. We study their well-posedness properties in $H^{s_c}(\mathbb{T}^3)$, where $\mathbb{T}^3$ is the three dimensional torus and $s_c = 3/2 - 1/p$ is the scaling-critical regularity. The convergence of solutions of the Hartree equation to solutions of the nonlinear Schr\"odinger equation is proved. We also consider the case of mixed nonlinearities, proving local well-posedness in $s_c$ by considering the problem as a perturbation of the higher-order Hartree equation. In the particular case of the (defocusing) quintic-cubic Hartree equation, we also prove global well-posedness for all initial conditions in $H^1(\mathbb{T}^3)$. This is done by viewing it as a perturbation of the local quintic NLS.

math.AP↗

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↗

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↗