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Semiclassical Moser-Trudinger inequalities

Rakesh Arora, Phan Thành Nam, Phuoc-Tai Nguyen

Abstract

We extend the Moser-Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schrödinger operators on bounded domains.

Semiclassical Moser-Trudinger inequalities

Abstract

We extend the Moser-Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schrödinger operators on bounded domains.
Paper Structure (8 sections, 9 theorems, 117 equations)

This paper contains 8 sections, 9 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof strategy
  4. Proof of Theorem \ref{['thm:main1']}
  5. Proof of Theorem \ref{['thm:main-Sch']}
  6. Semiclassical Moser--Trudinger inequality for all dimensions
  7. Resolvent estimate
  8. Conclusion of Theorem \ref{['thm:main-Sch']}

Key Result

Theorem 1.1

Let $\Omega \subset { {\mathbb R} }^2$ be an open bounded set. Let $\{u_n\}_{n=1}^N \subset H_0^1(\Omega)$ satisfy the orthonormality assumpt:ON1. Define $\rho(x)=\sum_{n=1}^N |u_n(x)|^2$. Then for all $N\geqslant 1$ and all $\varepsilon\in (0,1/4]$ we have Consequently, for any $\alpha>0$ and $N\geqslant \max\{ e^4, e^{\frac{\alpha}{4\pi} +1} \}$, there holds

Theorems & Definitions (17)

  • Theorem 1.1: Semiclassical Moser--Trudinger inequality
  • Corollary 1.2
  • Theorem 1.3: Semiclassical bound for Schrödinger operators
  • Lemma 2.1: Uniform bound of one-body density with momentum cut-off
  • proof
  • proof : Proof of Theorem \ref{['thm:main1']}
  • Theorem 3.1: Semiclassical Moser--Trudinger inequality for all $d$
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 7 more