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Orthonormal Strichartz estimates on torus and waveguide manifold and applications

Divyang G. Bhimani, Subhash. R. Choudhary

TL;DR

This work develops orthonormal Strichartz estimates for fractional Schrödinger equations on the torus and waveguide manifolds, introducing sharp kernel bounds and novel decoupling for degeneracy-type surfaces. The authors extend the torus results of Nakamura and establish new waveguide results, connecting dispersive kernel theory with $ ext{ell}^2$-decoupling in degenerate regimes to obtain robust OSE and Strichartz bounds. These analytical tools feed into a Hartree model with infinitely many particles, yielding local well-posedness and small-data global well-posedness for non-trace-class initial data in both torus and waveguide settings. The results provide a framework for mean-field dynamics of fermionic systems on compact and hybrid geometries, with potential impact on quantum many-body analysis and dispersive PDE on manifolds.

Abstract

We establish new orthonormal Strichartz estimates for the fractional Schrödinger equations on torus $\mathbb T$ and waveguide manifold $\mathbb R^n\times \mathbb T^m$. We generalizes the result of Nakamura [42] on torus; while this is the first result on the waveguide manifold. The main novelty in this paper is the derivation of various kernel estimates associated to the fractional Schrödinger equations. Our kernel estimate generalizes the classical dispersive estimate on torus due to Kenig-Ponce-Vega [35]. On the other hand, we obtain new $\ell^2$ decoupling inequality for degeneracy type surfaces to treat the case of waveguide manifold; which maybe of independent interest and complements several known results. As an application, we establish local and small data global well-posednes for the Hartree equation with infinitely many particles with non-trace class initial data.

Orthonormal Strichartz estimates on torus and waveguide manifold and applications

TL;DR

This work develops orthonormal Strichartz estimates for fractional Schrödinger equations on the torus and waveguide manifolds, introducing sharp kernel bounds and novel decoupling for degeneracy-type surfaces. The authors extend the torus results of Nakamura and establish new waveguide results, connecting dispersive kernel theory with -decoupling in degenerate regimes to obtain robust OSE and Strichartz bounds. These analytical tools feed into a Hartree model with infinitely many particles, yielding local well-posedness and small-data global well-posedness for non-trace-class initial data in both torus and waveguide settings. The results provide a framework for mean-field dynamics of fermionic systems on compact and hybrid geometries, with potential impact on quantum many-body analysis and dispersive PDE on manifolds.

Abstract

We establish new orthonormal Strichartz estimates for the fractional Schrödinger equations on torus and waveguide manifold . We generalizes the result of Nakamura [42] on torus; while this is the first result on the waveguide manifold. The main novelty in this paper is the derivation of various kernel estimates associated to the fractional Schrödinger equations. Our kernel estimate generalizes the classical dispersive estimate on torus due to Kenig-Ponce-Vega [35]. On the other hand, we obtain new decoupling inequality for degeneracy type surfaces to treat the case of waveguide manifold; which maybe of independent interest and complements several known results. As an application, we establish local and small data global well-posednes for the Hartree equation with infinitely many particles with non-trace class initial data.

Paper Structure

This paper contains 28 sections, 30 theorems, 311 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Classical Strichartz estimates
  3. Known orthonormal Strichartz estimates
  4. Statement of the main results
  5. OSE on torus
  6. Decoupling inequalities and fractional Strichatz estimates on waveguide manifold
  7. Orthonormal Strichatz estimates on waveguide manifold
  8. Hartree Eqs. with infinitely many particles
  9. Motivation and description of the problem
  10. Application to well-posedness
  11. Preliminaries
  12. Notations
  13. Besov spaces
  14. Schatten spaces $\mathfrak{S}^{\alpha}$
  15. Duality principle in $\mathfrak{S}^{\alpha}$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Assume that for $\mathcal{M}= \mathbb T^d$ or $\mathbb R^n \times \mathbb T^m$ with $d=n+m,$ Then, for any $\epsilon >0$ and $N>1,$ we have

Figures (3)

  • Figure 1: The line joining $B$ to $C$ represents equation \ref{['adc']} for $d>2.$ The point $A$ is a critical point. The segment $BA$ represents subcritical regime. The segment $AC$ represents supercritical regime.
  • Figure 2: The points $A$ to $B$ in one dimension for $\theta >3.$
  • Figure 3: Case $n=m=1$: red = $\mathrm{I}$, green = $\mathrm{II}$, yellow = $\mathrm{III}$, white = $\mathrm{IV}$.

Theorems & Definitions (56)

  • Theorem 1.1: bourgain2015BourgainrestrictionBarronAlex
  • Theorem 1.2: Dinh
  • Theorem 1.3: OSE on $\mathbb R^d$ frank2014strichartzfrank2016steinneal2019
  • Theorem 1.4: OSE on $\mathbb {T}^{d}$ nakamura2020orthonormal
  • Theorem 1.5: fractional OSE on $\mathbb T^d$ wang2025strichartz
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 46 more