[Paper Review] Resonances for 1D half-line periodic operators: I. Generic case

Trinh Tuan Phong|arXiv (Cornell University)|Sep 13, 2015

Spectral Theory in Mathematical Physics4 references3 citations

TL;DR

This paper analyzes resonances of one-dimensional half-line Schrödinger operators with truncated periodic potentials in the generic case, where the spectral parameters near the essential spectrum boundary decay as $|\lambda_k - E_0|/L$. Using a refined asymptotic analysis of the resonance equation involving the meromorphic continuation of the resolvent, it proves the absence of resonances in a specific region near the spectral edge, establishing a free resonance region below compact spectral intervals. The key result is the localization of resonances away from the boundary of the essential spectrum under generic conditions.

ABSTRACT

The present paper addresses questions on resonances for a $1$D Schrödinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-Δ+V$ and $H^{\mathbb N}_L = -Δ+ V 1_{[0,L]}$ acting on $\ell^{2}(\mathbb N)$ with Dirichlet boundary condition at $0$ with $L \in \mathbb N$. We describe the resonances of $H^{\mathbb N}_{L}$ near the boundary of the essential spectrum of $H^{\mathbb N}$ as $L ightarrow +\infty$ in the generic case.

Motivation & Objective

• To understand the distribution of resonances for truncated periodic Schrödinger operators on the half-line as the truncation length $L \to \infty$.
• To resolve the behavior of resonances near the boundary of the essential spectrum $\partial\Sigma_{\mathbb{Z}}$ when the spectral parameters exhibit generic decay $a_k \asymp |\lambda_k - E_0|/L$.
• To extend previous results that required resonances to be bounded away from $\partial\Sigma_{\mathbb{Z}}$ by analyzing the critical regime near the spectral edge.
• To establish the existence of a free resonance region—free of resonances—just below compact intervals in the essential spectrum, under generic conditions on the potential.

Proposed method

• The resonance equation $S_L(E) = \sum_{k=0}^L \frac{a_k}{\lambda_k - E} = -e^{-i\theta(E)}$ is analyzed, with $E = 2\cos\theta(E)$, where $\theta(E)$ is analytically continued to the lower half-plane.
• The imaginary part of $S_L(E)$ is used to derive $\text{Im}S_L(E) = \text{Im}E \sum_{k=0}^L \frac{a_k}{|\lambda_k - E|^2} = e^{\text{Im}E} \sin(\text{Re}\theta(E))$, which is positive in the lower half-plane.
• A harmonic function argument is applied to $\text{Im}S_L(E)$ in a rectangular region $\mathcal{R}_1$ near $E_0 \in \partial\Sigma_{\mathbb{Z}}$, using the maximum principle to bound the imaginary part.
• The sum $\sum_{k=0}^{\varepsilon L} \frac{a_k y}{(\lambda_k - x)^2 + y^2}$ is estimated on the boundary of $\mathcal{R}_1$ by splitting into low- and high-mode regimes based on $k \lesssim \varepsilon^2 L$ and $k \gtrsim \varepsilon^2 L$.
• Asymptotic estimates for $a_k \asymp |\lambda_k - E_0|/L \asymp k^2/L^3$ are used to bound the sum uniformly in the region, showing $|\text{Im}S_L(E)| \lesssim \varepsilon$.
• The proof relies on the generic decay of $a_k$, which allows control of the sum near the spectral edge, contrasting with the non-generic case where $a_k \asymp 1/L$.

Experimental results

Research questions

• RQ1 What is the distribution of resonances for half-line periodic Schrödinger operators near the boundary of the essential spectrum when the potential is in the generic case?
• RQ2 Can resonances accumulate near $\partial\Sigma_{\mathbb{Z}}$ for $H_L^{\mathbb{N}}$ as $L \to \infty$ under generic decay assumptions on the spectral parameters $a_k$?
• RQ3 Is there a free resonance region—i.e., a region without resonances—just below compact intervals in the essential spectrum for such operators?
• RQ4 How does the behavior of $a_k = |\varphi_k(L)|^2$ near the spectral edge affect the resonance distribution, particularly in the generic case $a_k \asymp |\lambda_k - E_0|/L$?

Key findings

• There are no resonances in the region $\mathcal{R}_1 = [E_0 - \varepsilon, E_0] - i[\frac{C_0}{L^2}, \varepsilon^5]$ for sufficiently large $C_0 > 0$, under the generic decay condition $a_k \asymp |\lambda_k - E_0|/L$.
• The imaginary part of the spectral sum $\text{Im}S_L(E)$ is bounded by $\varepsilon$ in $\mathcal{R}_1$, implying no poles of the meromorphic continuation of the resolvent exist there.
• The absence of resonances is established via a harmonic function argument on the boundary of $\mathcal{R}_1$, using the maximum principle to control $\text{Im}S_L(E)$.
• The estimate $S \lesssim \varepsilon$ holds uniformly on the boundary of $\mathcal{R}_1$, achieved by splitting the sum into low- and high-mode contributions and using $a_k \asymp k^2/L^3$.
• The result confirms that in the generic case, resonances do not accumulate near $\partial\Sigma_{\mathbb{Z}}$ in the region just below compact spectral intervals, even as $L \to \infty$.
• The analysis shows that the generic decay $a_k \asymp |\lambda_k - E_0|/L$ leads to a significant suppression of resonance density near the spectral edge, compared to the non-generic case.