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

Tuan Phong Trinh|arXiv (Cornell University)|Sep 21, 2015

Spectral Theory in Mathematical Physics4 references3 citations

TL;DR

This paper studies resonances of one-dimensional half-line Schrödinger operators with truncated periodic potentials, focusing on the non-generic case where the spectral parameter $ a_k \asymp 1/L $ near the band edge $ E_0 \in \partial\Sigma_{\mathbb{Z}} $. Using a complex analytic approach involving a transformed resonance equation and contour deformation, the authors prove that no resonances exist in a neighborhood of $ E_0 $ as $ L \to \infty $, establishing a sharp absence result in the non-generic regime.

ABSTRACT

The present paper is devoted to the study of resonances for a $1$D Schrödinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-Δ+V$ and $H^{\mathbb N}_{L}= -Δ+ V1_{[0, L]}$ 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 $L ightarrow +\infty$ under a special assumption.\\ The present paper is in a series of our research papers on resonances

Motivation & Objective

• To analyze the distribution of resonances for truncated 1D periodic Schrödinger operators $ H^\mathbb{N}_L $ near the boundary of the essential spectrum $ \partial\Sigma_{\mathbb{Z}} $.
• To address the case where the spectral parameters $ a_k \asymp 1/L $, which is non-generic and previously inaccessible with standard methods.
• To establish the absence of resonances in a neighborhood of the band edge $ E_0 \in \partial\Sigma_{\mathbb{Z}} $ under this non-generic condition.
• To develop a refined complex analytic technique to handle the resonance equation near spectral boundaries where standard methods fail.

Proposed method

• Transform the resonance equation $ S_L(E) = -e^{-i\theta(E)} $ into a new function $ f_L(z) $ via a change of variables $ z = x + iy $, $ y \asymp -1/\varepsilon L $, to analyze behavior near $ E_0 $.
• Use a spectral transformation $ \tilde{\lambda}_k = L^2(\lambda_k - E_0) $, $ \tilde{a}_k = L^2 a_k $ to rescale eigenvalues and weights near the band edge.
• Apply contour deformation to isolate the domain $ \Omega^i $ around $ E_0 $, and prove that $ f_L $ is univalent (injective) and bounded away from zero in this region.
• Establish lower bounds on $ |f_L'(z)| \gtrsim 1 $ and $ \text{Re}f_L(z) \asymp 1 $, ensuring $ f_L $ maps $ \Omega^i $ to a region avoiding $ -e^{-i\theta(E_0)}/L $.
• Use asymptotic estimates on $ \sum \tilde{a}_k / (\tilde{\lambda}_k - x)^2 + y^2 \asymp 1 $ to control the imaginary part and ensure no poles in $ \Omega^i $.
• Conclude that since $ -e^{-i\theta(E_0)}/L \notin f_L(\Omega^i) $, the resonance equation has no solution in $ \Omega^i $, implying no resonances exist near $ E_0 $.

Experimental results

Research questions

• RQ1 Do resonances of $ H^\mathbb{N}_L $ exist near the band edge $ E_0 \in \partial\Sigma_{\mathbb{Z}} $ when $ a_k \asymp 1/L $?
• RQ2 Can the standard resonance analysis techniques be extended to the non-generic case $ a_k \asymp 1/L $ near spectral boundaries?
• RQ3 What is the asymptotic behavior of resonances in the limit $ L \to \infty $ when the spectral weights $ a_k $ are not exponentially small near $ E_0 $?
• RQ4 Is it possible to prove the absence of resonances in a neighborhood of $ E_0 $ under the non-generic $ a_k \asymp 1/L $ condition?

Key findings

• No resonances exist in the domain $ \Omega^i = [E_0, E_0 + \varepsilon_1] - i[0, \varepsilon_2] $ for sufficiently large $ L $, under the non-generic assumption $ a_k \asymp 1/L $.
• The transformed function $ f_L(z) $ is univalent on $ \Omega^i $, with $ |f_L'(z)| \gtrsim 1 $ and $ \text{Re}f_L(z) \asymp 1 $, ensuring no solutions to the resonance equation in this region.
• The image of $ \Omega^i $ under $ f_L $ stays uniformly bounded away from zero, so $ -e^{-i\theta(E_0)}/L \notin f_L(\Omega^i) $, implying no resonances in $ \Omega^i $.
• The asymptotic estimate $ \sum_{k=0}^L \frac{\tilde{a}_k}{(\tilde{\lambda}_k - x)^2 + y^2} \asymp 1 $ holds uniformly in $ \Omega^i $, controlling the behavior of the resonance equation.
• The result holds under the condition $ \varepsilon_1 \asymp \varepsilon^2 $, ensuring the domain is small enough to allow uniform estimates.