Coupling in the singular limit of thin quantum waveguides

Graph-like models for thin waveguides with Robin boundary conditions

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that the projections on each transverse mode generically give rise to decoupling conditions between the edges of the graph while exceptionally a coupling can occur. The non decoupling conditions are related to the existence of resonances at the thresholds of the continuum spectrum.

On the spectrum of narrow Neumann waveguide with periodically distributed $δ'$ traps

We analyze a family of singular Schrödinger operators describing a Neumann waveguide with a periodic array of singular traps of a $δ'$ type. We show that in the limit when perpendicular size of the guide tends to zero and the $δ'$ interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.

Discrete spectrum of thin PT-symmetric waveguide

In a thin multidimensional layer we consider a second order differential PT-symmetric operator. The operator is of rather general form and its coefficients are arbitrary functions depending both on slow and fast variables. The PT-symmetry of the operator is ensured by the boundary conditions of Robin type with pure imaginary coefficient. In the work we determine the limiting operator, prove the uniform resolvent convergence of the perturbed operator to the limiting one, and derive the estimates for the rates of convergence. We establish the convergence of the spectrum of perturbed operator to …

Spectral asymptotics for two-dimensional Dirac operators in thin waveguides

We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $κ$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schrödinger operator on $L^2(\mathbb R)$ \[ \mathcal{L}_e := -\frac{d^2}{ds^2} - \frac{κ^2}{π^2} \] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $\varepsilon$ from the essential spectrum, where $2\varepsilon$ is the width of the waveguide. This is in contr…

Klein-Gordon奇异波导的积分公式化

Integral formulation of Klein-Gordon singular waveguides

针对二维分隔绝缘相的奇异波导，本文建立积分公式并分析其性质，采用快速多极与扫描加速迭代法求解，数值验证快速收敛及散射效应。

连续谱中Friedrich-Wintgen束缚态的存在性：带有细波导开口的谐振腔

Existence of Friedrich-Wintgen Bound States in the Continuum: Cavity with a Thin Waveguide Opening

本文研究了二维电磁谐振腔通过细波导耦合时Friedrich-Wintgen束缚态的存在性，通过引入折射率扰动并证明在特定条件下束缚态的交点存在。

A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens.

Thin waveguides with Robin boundary conditions

We consider the Laplace operator in a thin three dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case, we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rota…

Absolute continuity of the spectrum in a twisted Dirichlet-Neumann waveguide

Quantum waveguide with the shape of planar infinite straight strip and combined Dirichlet and Neumann boundary conditions on the opposite half-lines of the boundary is considered. The absence of the point as well as of the singular continuous spectrum is proved.

A note on the spectrum of the Neumann Laplacian in periodic waveguides

We study the Neumann Laplacian $-Δ^N$ restricted to a periodic waveguide. In this situation its spectrum $σ(-Δ^N)$ presents a band structure. Our goal and strategy is to get spectral information from an analysis of the asymptotic behavior of these bands provided that the waveguide is sufficiently thin.

Absolute continuity and band gaps of the spectrum of the Dirichlet Laplacian in periodic waveguides

Consider the Dirichlet Laplacian operator $-Δ^D$ in a periodic waveguide $Ω$. On the condition that $Ω$ is sufficiently thin, we show that its spectrum $σ(-Δ^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $σ(-Δ^D)$ and locate it.

狄拉克奇异波导的积分公式

Integral formulation of Dirac singular waveguides

本文研究了二维狄拉克方程的边界积分公式，探讨了质量项在界面处的跃变对表面波的影响，并提出了一种快速数值方法进行求解。

Asymptotical study of two-layered discrete waveguide with a weak coupling

A thin two-layered waveguide is considered. The governing equations for this waveguide is a matrix Klein--Gordon equation of dimension~2. A formal solution of this system in the form of a double integral can be obtained by using Fourier transformation. Then, the double integral can be reduced to a single integral with the help of residue integration with respect to the time frequency. However, such an integral can be difficult to estimate since it involves branching and oscillating functions. This integral is studied asymptotically. A zone diagram technique is proposed to represent the set of …

The NLS limit for bosons in a quantum waveguide

We consider a system of $N$ bosons confined to a thin waveguide, i.e.\ to a region of space within an $\varepsilon$-tube around a curve in $\mathbb{R}^3$. We show that when taking simultaneously the NLS limit $N\to \infty$ and the limit of strong confinement $\varepsilon\to 0$, the time-evolution of such a system starting in a state close to a Bose-Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross-Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the "bend…