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Probabilistic Strichartz estimates in Schatten classes and their applications to the Hartree equation

Sonae Hadama, Takuto Yamamoto

TL;DR

The paper develops probabilistic Strichartz estimates in Schatten classes by randomizing initial operators, enabling control in strictly larger Schatten exponents than the deterministic bounds. Two randomization schemes are analyzed: a singular-value randomization and a full Wiener randomization, each yielding explicit $L^r$-type bounds for the density $\rho$ that surpass classical results, while preserving Schatten-class membership. These probabilistic Strichartz estimates are then applied to the Hartree equation for infinitely many particles, establishing local well-posedness in wider Schatten classes across dimensions and obtaining scattering results for the linearized equation. The approach avoids heavy reliance on orthogonality, uses a contraction-mapping framework based on a key bilinear estimate, and leverages recent orthonormal Strichartz improvements to treat nonlinear evolution in a probabilistic setting. The results broaden the functional-analytic landscape for many-particle quantum dynamics and offer new avenues for global behavior through probabilistic regularization.

Abstract

In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates. After that, many authors have studied this type of inequality. In this paper, we introduce two randomizations of operators and show that they allow us to treat strictly bigger Schatten exponents than the sharp exponents of the deterministic orthonormal Strichartz estimates. In the proofs, the orthogonality does not have any essential role, and randomness works instead of it. We also prove that our randomizations of operators never change the Schatten classes to which they originally belong. Moreover, we give some applications of our results to the Hartree and linearized Hartree equations for infinitely many particles. First, we construct local solutions to the Hartree equation with initial data in wide Schatten classes. By only using deterministic orthonormal Strichartz estimates, it is impossible to give any solution in our settings. Next, we consider the scattering problem of the linearized Hartree equation. One of our randomizations allows us to treat wide Schatten classes with some Sobolev regularities, and by the other randomization, we can remove all Sobolev regularities.

Probabilistic Strichartz estimates in Schatten classes and their applications to the Hartree equation

TL;DR

The paper develops probabilistic Strichartz estimates in Schatten classes by randomizing initial operators, enabling control in strictly larger Schatten exponents than the deterministic bounds. Two randomization schemes are analyzed: a singular-value randomization and a full Wiener randomization, each yielding explicit -type bounds for the density that surpass classical results, while preserving Schatten-class membership. These probabilistic Strichartz estimates are then applied to the Hartree equation for infinitely many particles, establishing local well-posedness in wider Schatten classes across dimensions and obtaining scattering results for the linearized equation. The approach avoids heavy reliance on orthogonality, uses a contraction-mapping framework based on a key bilinear estimate, and leverages recent orthonormal Strichartz improvements to treat nonlinear evolution in a probabilistic setting. The results broaden the functional-analytic landscape for many-particle quantum dynamics and offer new avenues for global behavior through probabilistic regularization.

Abstract

In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates. After that, many authors have studied this type of inequality. In this paper, we introduce two randomizations of operators and show that they allow us to treat strictly bigger Schatten exponents than the sharp exponents of the deterministic orthonormal Strichartz estimates. In the proofs, the orthogonality does not have any essential role, and randomness works instead of it. We also prove that our randomizations of operators never change the Schatten classes to which they originally belong. Moreover, we give some applications of our results to the Hartree and linearized Hartree equations for infinitely many particles. First, we construct local solutions to the Hartree equation with initial data in wide Schatten classes. By only using deterministic orthonormal Strichartz estimates, it is impossible to give any solution in our settings. Next, we consider the scattering problem of the linearized Hartree equation. One of our randomizations allows us to treat wide Schatten classes with some Sobolev regularities, and by the other randomization, we can remove all Sobolev regularities.
Paper Structure (42 sections, 25 theorems, 141 equations, 1 figure, 1 table)

This paper contains 42 sections, 25 theorems, 141 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Strichartz estimates in Schatten classes
  3. Hartree equation for infinitely many particles
  4. Randomization for compact operators
  5. Notations
  6. Main results
  7. Strichartz estimates in Schatten classes
  8. Applications to the Hartree equation for infinitely many particles
  9. Strategy of the proof
  10. Probabilistic Strichartz estimates
  11. Basic tools
  12. Proof of Theorem \ref{['prop:a.s.linear']}
  13. Proof of Theorem \ref{['prop:a.s.linear2']}
  14. No gain of regularity by the randomizations
  15. Application to the Hartree equation: Unique existence of the local solution
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $d \ge 1$. Let $p,q,\alpha \in [1,\infty]$ satisfy Then eq:SS and eq:OS hold for $\alpha = \frac{2q}{q+1}$. Moreover, $\alpha = \frac{2q}{q+1}$ is sharp in the following sense: If $\alpha > \frac{2q}{q+1}$, then eq:SS and eq:OS fail.

Figures (1)

  • Figure 1:

Theorems & Definitions (44)

  • Theorem 1.1: 2014FLLSTheorem 1; 2017FSTheorem 8
  • Proposition 1.2
  • Definition 1.3: Singular value randomization
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 34 more