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The $L^p$-boundedness of wave operators for 4-th order Schrödinger operators on $\mathbb{R}^2$, I

Artbazar Galtbayar, Kenji Yajima

TL;DR

This work addresses the problem of $L^p$-boundedness for the high-energy components of wave operators associated with the two-dimensional fourth-order Schrödinger operator $H = \Delta^2 + V(x)$. It develops a resolvent-centered, stationary framework and establishes good-operator (GOP) bounds for the high-energy part via a controlled Born-series analysis in two regimes: (i) small potentials, where the series converges in ${\bf B}(L^p)$, and (ii) larger potentials with rapid decay and no positive eigenvalues, where a detailed decomposition of the scattering matrix yields GOP for the high-energy wave operator. The results reduce the mapping properties of $f(H)P_{ac}(H)$ at high energy to those of the free multiplier and set the stage for low-energy analysis in a companion paper. Overall, the paper extends $L^p$-boundedness theory of wave operators to the 2D, higher-order setting and contributes essential resolvent-analytic tools and kernel estimates for future dispersive and scattering investigations.

Abstract

We prove that high energy parts of wave operators for fourth order Schrödinger operators $H=Δ^2 + V(x)$ in $\mathbb{R}^2$ are bounded in $L^p(\mathbb{R}^2)$ for $p\in(1,\infty)$.

The $L^p$-boundedness of wave operators for 4-th order Schrödinger operators on $\mathbb{R}^2$, I

TL;DR

This work addresses the problem of -boundedness for the high-energy components of wave operators associated with the two-dimensional fourth-order Schrödinger operator . It develops a resolvent-centered, stationary framework and establishes good-operator (GOP) bounds for the high-energy part via a controlled Born-series analysis in two regimes: (i) small potentials, where the series converges in , and (ii) larger potentials with rapid decay and no positive eigenvalues, where a detailed decomposition of the scattering matrix yields GOP for the high-energy wave operator. The results reduce the mapping properties of at high energy to those of the free multiplier and set the stage for low-energy analysis in a companion paper. Overall, the paper extends -boundedness theory of wave operators to the 2D, higher-order setting and contributes essential resolvent-analytic tools and kernel estimates for future dispersive and scattering investigations.

Abstract

We prove that high energy parts of wave operators for fourth order Schrödinger operators in are bounded in for .

Paper Structure

This paper contains 7 sections, 12 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Resolvents
  3. Preliminaries
  4. Proof of Lemma \ref{['lm:H02y']}.
  5. Proof of Theorems
  6. Proof Theorem \ref{['th:small']}
  7. Proof Theorem \ref{['th:high']}

Key Result

Theorem 1.2

Suppose $M_q(V) \in L^1({\mathbb R}^2)$ for a $q>1$ and $\langle \log |x|\rangle^{2} V \in L^1({\mathbb R}^2)$. Let $a>0$. Then, there exists $c_0>0$ such that $W_{\pm} \chi_{\geq{a}}(|D|)$ are GOP whenever $\|V\|_{L^{q}_{loc,u}}+ \|\langle \log |x|\rangle^{2} V\|_{L^1}\leq c_0$.

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 10 more