周期量子波导中由边界测量稳定确定不定常标量势

在无限波导中确定相对论波动方程中与时间无关势的Hölder稳定性估计

Hölder stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide

本文针对无限波导几何中依赖三个与时间无关势的相对论波动方程，建立了Calderón问题的Hölder稳定性估计，并明确确定了常数和Hölder指数对Sobolev指数的依赖关系。

Bands in the spectrum of a periodic elastic waveguide

We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $ h >0$. The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as $h \to 0$.

On the generation of stable Kerr frequency combs in the Lugiato-Lefever model of periodic optical waveguides

We consider the Lugiato-Lefever (LL) model of optical fibers. We construct a two parameter family of steady state solutions, i.e. Kerr frequency combs, for small pumping parameter $h>0$ and the correspondingly (and necessarily) small detuning parameter, $α>0$. These are $O(1)$ waves, as they are constructed as bifurcation from the standard cnoidal solutions of the cubic NLS. We identify the spectrally stable ones, and more precisely, we show that the spectrum of the linearized operator contains the eigenvalues $0, -2α$, while the rest of it is a subset of $ \{μ: \Reμ=-α\}$. This is in line wit…

Spectrum decomposition of translation operators in periodic waveguide

Scattering problems in periodic waveguides are interesting but also challenging topics in mathematics, both theoretically and numerically. Due to the existence of eigenvalues, the unique solvability of these problems is not always guaranteed. To obtain a unique solution that is "physically meaningful", the limiting absorption principle (LAP) is a commonly used method. LAP assumes that the limit of a family of solutions with absorbing media converges, as the absorption parameter tends to 0, and the limit is the "physically meaningful solution". It is also called the LAP solution in this paper. …

Uniqueness and stability results for an inverse spectral problem in a periodic waveguide

Let $Ω=ω\times\mathbb R$ where $ω\subset \mathbb R^2$ be a bounded domain, and $V : Ω\to\mathbb R$ a bounded potential which is $2π$-periodic in the variable $x_{3}\in \mathbb R$. We study the inverse problem consisting in the determination of $V$, through the boundary spectral data of the operator $u\mapsto Au := -Δu + Vu$, acting on $L^2(ω\times(0,2π))$, with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials $V_{1}$ and $V_{2}$ we denote by $(λ_{1,k})_{k}$ and $(λ_{2,k})_{k}$ the eigenvalues associated to the operators $A_{1}$ and $A_{2}$ (th…

Example of periodic Neumann waveguide with gap in spectrum

In this note we investigate spectral properties of a periodic waveguide $Ω^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon=-ρ^\varepsilonΔ_{Ω^\varepsilon}$, where $Δ_{Ω^\varepsilon}$ is the Neumann Laplacian in $Ω^\varepsilon$, the weight $ρ^\varepsilon$ is equal to $1$ everywhere except the union of the "rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilo…

小窗口周期系统耦合波导中的带隙开口和分裂带边

Gap opening and split band edges in waveguides coupled by a periodic system of small windows

在两个耦合波导的例子中，我们构造了一个周期性的二阶微分算子，它作用在欧几里得整环中，并且具有严格在布里渊区内达到边缘的谱隙。在两个不同宽度的无限长的带，有一个共同的内部边界的拉普拉斯算子的波导建模。在这个共同的边界上，我们施加诺依曼边界条件，但削减了一个周期性的系统的小孔，而其余的外部边界上，我们施加狄利克雷边界条件。它示出，通过改变带的宽度和孔之间的距离，可以控制的带函数的极值的位置以及开放的间隙的数量。我们计算了间隙长度和极值位置的渐近性中的首项。

Spectrum of a singularly perturbed periodic thin waveguide

We consider a family $\{Ω^\varepsilon\}_{\varepsilon>0}$ of periodic domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-Δ_{Ω^\varepsilon}$ on $Ω^\varepsilon$. The waveguide $Ω^\varepsilon$ is a union of a thin straight strip of the width $\varepsilon$ and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period $\varepsilon$, along the strip upper boundary. For $\varepsilon\to 0$ we prove a (kind of) resolvent convergence of $-Δ_{Ω^\varepsilon}$ to a certa…

非周期交替Dirichlet和Robin条件波导的均匀化和渐近性

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

本文考虑了平面无限带中的磁Schroedinger算子，其Dirichlet和Robin边界条件是频繁和非周期交替的。假设齐次化边界条件是Dirichlet或Robin边界条件，我们建立了在各种算子范数下的一致预解式收敛性，并证明了收敛速度的估计.结果表明，这些估计可以通过使用特殊的边界校正得到改善。在周期交替，纯拉普拉斯算子，和均匀化罗宾边界条件的情况下，我们构造两项渐近的第一带函数，以及完整的渐近展开（指数小的长期）的底部的带谱。

开放周期波导局部扰动的散射问题的 PML 方法

The PML-Method for a Scattering Problem for a Local Perturbation of an Open Periodic Waveguide

本文研究了开放周期波导局部扰动波散射问题中完美匹配层（PML）方法的收敛性，基于 Floquet-Bloch 变换和极限吸收原理，通过比较 Dirichlet-to-Neumann 映射证明了 PML 方法呈指数收敛，并给出了数值算例验证。

Spectral gaps in a double-periodic perforated Neumann waveguide

We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter $\varepsilon>0$. The periodicity cell itself contains a string of holes at a distance $O(\varepsilon)$ between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as $\varepsilon \to 0$. We demonstrate that, for $\varepsilon$ small enough, some spectral gaps are open. The position…

从 Dirac 点分岔的波导中界面模式的数学理论

Mathematical theory for the interface mode in a waveguide bifurcated from a Dirac point

本文针对由同一 Dirac 点周期介质小幅扰动形成的双半无限结构波导，利用层势、渐近分析与边界积分算子证明共同带隙内存在从 Dirac 点分岔的指数衰减界面模式。

量子波导中的方向性弹道输运

Directional Ballistic Transport in Quantum Waveguides

本文研究了量子波导中周期性方向上的弹道输运现象，通过Floquet理论和Dirichlet到Neumann映射揭示了表面态的输运特性。