[Paper Review] $L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions

Kenji Yajima|arXiv (Cornell University)|Jun 17, 2020

Advanced Mathematical Physics Problems22 references4 citations

TL;DR

This paper establishes the $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators with point interactions. It proves that the wave operators are bounded on $L^p(\mathbb{R}^2)$ for all $1 < p < ∞$ if and only if there are no $p$-wave resonances; otherwise, they are bounded only for $1 < p \leq 2$ and unbounded for $2 < p < \infty$, with the result determined by the spectral structure of the interaction matrix.

ABSTRACT

For two dimensional Schrödinger operator $H$ with point interactions, We prove that wave operators of scattering for the pair $(H,H_0)$, $H_0$ being the free Schrödinger operator, are bounded in the Lebesgue space $L^p(\R^2)$ for $1

Motivation & Objective

• To determine the $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators with point interactions.
• To characterize the role of generalized eigenfunctions and resonances—specifically $s$-wave and $p$-wave resonances—in determining the $L^p$-boundedness of wave operators.
• To establish a sharp dichotomy in boundedness behavior based on the presence or absence of $p$-wave resonances in the spectrum of the operator.

Proposed method

• The study uses the resolvent construction of the Schrödinger operator $H_{\alpha,Y}$ via the Green's function and the Hankel function $H_0^{(1)}$, which governs the point interaction's behavior.
• It defines the operator-valued resolvent $R(z^2)$ through a perturbation of the free resolvent using an $N \times N$ matrix $\Gamma_{\alpha,Y}(z)$, which encodes the interaction strengths and spatial configuration.
• The wave operators $W_{\alpha,Y}^\pm$ are defined as strong limits in $L^2(\mathbb{R}^2)$, and their $L^p$-boundedness is analyzed via asymptotic expansions of $\Gamma_{\alpha,Y}(\lambda)$ as $\lambda \to 0^+$.
• The analysis relies on the asymptotic behavior of the Green's function and the spectral properties of three symmetric matrices: $\tilde{\mathcal{D}}$, $\mathcal{G}_1(Y)$, and $\mathcal{G}_2(Y)$, which govern the low-energy behavior of the system.
• The proof employs microlocal and Fourier analysis techniques, including the use of cutoff functions $\chi_{\leq 2\varepsilon}$ and estimates on oscillatory integrals involving $|\xi| + |p|$.
• A key technical tool is the use of a modified version of the Kato-Rosenblum theorem and the construction of good operators via spectral projections and matrix inversion in the low-energy limit.

Experimental results

Research questions

• RQ1 Under what conditions are the wave operators for 2D Schrödinger operators with point interactions bounded in $L^p(\mathbb{R}^2)$ for $1 < p < \infty$?
• RQ2 How do the presence or absence of $p$-wave resonances affect the $L^p$-boundedness of wave operators?
• RQ3 What is the precise threshold behavior of the wave operators when $p > 2$, and why do they become unbounded?

Key findings

• The wave operators are bounded in $L^p(\mathbb{R}^2)$ for all $1 < p < \infty$ if and only if there are no $p$-wave resonances in the spectrum of $H_{\alpha,Y}$.
• If $p$-wave resonances are present, the wave operators remain bounded for $1 < p \leq 2$ but become unbounded for $2 < p < \infty$, indicating a sharp transition at $p = 2$.
• The absence of $p$-wave resonances is equivalent to the non-degeneracy of the matrix $T_e^\perp G_1 T_e^\perp$ in the low-energy expansion, which ensures the invertibility of the relevant operator in the wave operator construction.
• The presence of $s$-wave resonances (nonzero $a$) does not obstruct $L^p$-boundedness for $1 < p < \infty$, but $p$-wave resonances ($a=0$, $b \neq 0$) do.
• The boundedness failure for $2 < p < \infty$ arises from the non-integrability of certain Fourier multiplier components related to the $p$-wave contribution, as shown via $L^p$-norm estimates on oscillatory integrals.
• A technical lemma (Lemma A) provides a pointwise decay estimate for a double Fourier integral involving cutoffs, which is essential for controlling the operator norm in the $L^p$-boundedness proof.