Schroedinger operators on exterior domains with Robin boundary conditions: heat kernel estimates

Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (arXiv:1608.04896, to appear in J. Convex Anal.), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that unde…

超平面外域中最低Robin本征值的优化

Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane

研究超平面外域中Robin本征值的优化问题，证明测地盘外的最低本征值在固定面积或周长约束下最大，且获得临界常数的反向不等式。

外区域中第一Robin特征值的几何优化

Geometric Optimization of the First Robin Eigenvalue in Exterior Domains

该研究将二维下球使离散第一Robin特征值最大化的结论推广到高维，证明近球形区域中球是局部极大值点，但一般并非全局极大值点。

Remark on magnetic Schroedinger operators in exterior domains

We study the Schroedinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which appears to be independent from the boundary condition.

有界区域和外区域中吸收与生成 $p$-Robin 特征值的行为

Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains

本文为广义 $p$-Robin 问题的第一特征值建立了严格定量不等式，统一推导了有界区域中所有 $p$ 及小参数 $α$ 的精确渐近行为，并刻画了外区域特征值存在性与渐近性，改进了现有结果。

On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon

Let $Ω\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $α>0$, let $H^Ω_α$ denote the Laplacian in $Ω$, $u\mapsto -Δu$, with the Robin boundary conditions $\partial u/\partialν=αu$, where $ν$ is the exterior unit normal at the boundary of $Ω$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^Ω_m(α)$ of $H^Ω_α$ behaves as \[ E^Ω_m(α)=-α^2+μ^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrtα}\Big) \quad {as $α$ tends to $+\infty$}, \] where $μ^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ de…

无界域上带点相互作用的Robin拉普拉斯算子：离散谱与耦合渐近行为

Robin Laplacians with point interactions on unbounded domains: discrete spectrum and coupling asymptotics

本文研究二维、三维无界域及外域上带有限多点相互作用的Neumann、Robin拉普拉斯算子的离散谱，推导本征值计数公式与渐近行为，通过外球、外盘模型阐明临界耦合特性。

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

We consider a magnetic Schroedinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for…

Spectral optimization for Robin Laplacian on domains admitting parallel coordinates

In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $Ω$ corresponding to a negative Robin parameter does not exceed the analogous quantity …

Eigenvalue asymptotics of the even-dimensional exterior Landau-Neumann Hamiltonian

We study the Schroedinger operator with a constant magnetic field in the exterior of a compact domain in $\mathbb{R}^{2d}$, $d\geq 1$. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhart domain we are able to show a more precise asymptotic formula.

On the discrete spectrum of Schroedinger operators with Ahlfors regular potentials in a strip

In this paper, quantitative upper estimates for the number of eigenvalues lying below the essential spectrum of Schroedinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L^1 norms and Orlicz norms of the potential.

半直线上非自伴离散薛定谔算子的谱包络与稳定性

Spectral enclosures and stability for non-self-adjoint discrete Schroedinger operators on the half-line

本文研究半直线上带复Robin边界和复值势的非自伴离散薛定谔算子，建立了可求和势的最优谱包络、谱稳定性的一般小性条件，并证明了一个可生成最优离散Hardy不等式的一般恒等式。

Curvature-induced bound states in Robin waveguides and their asymptotical properties

We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive …

平面区域内小目标的竞争：从Dirichlet边界条件到Robin与Steklov边界条件

Competition of small targets in planar domains: from Dirichlet to Robin and Steklov boundary condition

本文采用匹配渐近展开法，研究平面区域内多小目标对扩散粒子的竞争效应，推导不同边界条件下的渐近性质，结果可应用于扩散控制反应领域。

Clusters of eigenvalues for the magnetic Laplacian with Robin condition

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.

Operator estimates for homogenization of the Robin Laplacian in a perforated domain

Let $\varepsilon>0$ be a small parameter. We consider the domain $Ω_\varepsilon:=Ω\setminus D_\varepsilon$, where $Ω$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $Δ_\varepsilon$ be the Laplace operator in $Ω_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+γ_\varepsilon u = 0$ with $γ_\varepsilon\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have sho…

Perturbation of essential spectra of exterior elliptic problems

For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it i…

Approximations of spectra of Schrödinger operators with complex potentials on $\mathbb{R}^d$

We study spectral approximations of Schrödinger operators $T=-Δ+Q$ with complex potentials on $Ω=\mathbb{R}^d$, or exterior domains $Ω\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $Ω_n$, and of boundary conditions on $\partial Ω_n$ such as mixed Dirichlet/Robin type. In particular, ${\rm Re} \, Q$ need not be bounded from below and $Q$ may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of $T$ by those …

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for th…