Michael Goldberg, Wilhelm Schlag
We prove a limiting absorption principle for linear Schroedinger equations in Lebesgue spaces. In particular, we do not require any polynomially decaying weights as in the classical Agmon estimate. The methods used are close to the Stein-Tomas theorem in harmonic analysis.
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The limiting absorption principle in two-dimensional space is justified for a second-order elliptic operators. Necessary and sufficient conditions for the right-hand side are given for this principle to be valid.
We provide a limiting absorption principle for self-adjoint realizations of Dirac operators with electrostatic and Lorentz scalar $δ$-shell interactions supported on regular compact surfaces. Then we show completeness of the wave operators and give a representation formula for the scattering matrix.
Limiting absorption principle for a hybrid resonance in a two-dimensional cold plasma
本文针对二维冷等离子体中混合共振的边值问题，通过加权Sobolev空间中的先验界证明极限吸收原理，并利用解的正则与奇异分解引入辐射条件，确立适定问题。
Limiting absorption principle of Helmholtz equation with sign changing coefficients under periodic structure
本文通过补余边界条件建立亥姆霍兹方程的严格先验估计，解析推导极限吸收原理，并证明二维周期结构中常规与负折射率材料界面电磁传输问题的适定性。
We prove the limiting absorption principle and discuss the continuity properties of the boundary values of the resolvent for a class of form bounded perturbations of the Euclidean Laplacian $Δ$ that covers both short and long range potentials with an essentially optimal behaviour at infinity.
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
Making use of the weighted Mourre theory developed in [GJ1], we show the limiting absorption principle for Schr{ö}dinger operators with perturbed oscillating potential on appropriate energy intervals. We focus on a certain class of oscillating potentials (larger than the one in [GJ2]) that was already studied in [BD, MU, ReT1, ReT2]. We allow long-range and short-range components and local singularities in the perturbation. We improve known results, the main novelty being the presence of a long-range perturbation. A subclass of the considered potentials actually cannot be treated by the usual …
We review the concept of the limiting absorption principle and its connection to virtual levels of operators in Banach spaces.
Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle
本文通过极限吸收原理解决一维准周期介质中频率域波传播的不稳定性问题，提出罗宾-罗宾边界条件并设计数值方法计算物理解。
We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.
In this paper, we shall show that the limiting absorption principle for the wave operator on the asymptotically Minkowski spacetime. This problem was previously considered by [A. Vasy, J. Spect. Theory, 10,439-461 , (2020)]. Here, we employ a more transparent tool, the Mourre theory and removes an additional condition which is imposed in his paper. Moreover, we also prove that the anti-Feynman propagator defined by Gérard and Wrochna coincides with the outgoing resolvent.
In this paper, we prove a limiting absorption principle for high-order Schrödinger operators with a large class of potentials which generalize some results by A. Ionescu and W. Schlag. Our main idea is to handle the boundary operators by the restriction theorem of Fourier transform. Two key tools we use in this paper are the Stein--Tomas theorem in Lorentz spaces and a sharp trace lemma given by S. Agmon and L. Hörmander
We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schroedinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We also weaken the regularity assumptions on the potential.
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{ö}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to relax conditions on the derivatives of the potential that were required up to now.
Limiting absorption principle and well-posedness for the time-harmonic Maxwell equations with anisotropic sign-changing coefficients
研究时谐域中具有各向异性变号系数的 Maxwell 方程，通过变量替换建立 Cauchy 问题并采用 Fourier 方法与变分方法推导先验估计，获得了具有局部特征且易于验证的系数条件，从而证明极限吸收原理与适定性，为负折射率超材料中电磁场稳定性提供充分判据。
Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. &(2) \quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0, where $k>0$ is a constant. Necessary and sufficient conditions are given for the operator $L$ not to have eigenvalues in the half-plane Re$z<0$ and not to have a positive eigenvalue at a given point $k_d^2 >0$. These conditions are given in terms of the large-time behavior of the s…
This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schrödinger operators $H=-Δ+V$, i.e., In $\mathbb{R}^n$ ($n\ge 3$) we prove the $ε-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-λ\pm iε)^{-1}$ for all $λ>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schrödinger operators $H$. Next, we consider the fractional Sch…
Schrödinger-Poisson systems with zero mass in the Sobolev limiting case
本文研究ℝ^N中Sobolev临界情形下零质量拟线性Schrödinger与高阶分数阶Poisson强耦合系统，借助变号对数Riesz核的Choquard方程及多项式核一致逼近的变分近似方法证明正解存在。
In this paper we consider Dirac operators in $\mathbb R^n$, $n\geq2$, with a potential $V$. Under mild decay and continuity assumptions on $V$ and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted $L^2$ spaces as the frequency goes to infinity.
We prove an $L^p$-version of the limiting absoprtion principle for a class of periodic elliptic differential operators of second order. The result is applied to the construction of nontrivial solutions of nonlinear Helmholtz equations with periodic coefficient functions.
We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the fre…
In this work we investigate the L^p-L^q-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an L^p-L^q-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.
Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
本文研究二维分数阶 Sobolev 极限情形下 Schrödinger-Poisson 系统的解的存在性与定性性质，通过辅助 Choquard 方程的变分逼近方法证明解存在，并利用移动平面技巧得到对称性与衰减结果。
Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces.
We study the following Helmholtz equation $$ (\nabla +iA(x))^{2} u+ V_{1}(x) u + V_{2}(x) u + λu = f(x) $$ in $\mathbb{R}^d$ with magnetic and electric potentials that are singular at the origin and decay at infinity. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.
We consider the semi-classical limit of nonlinear Schrodinger equations in the presence of both a polynomial nonlinearity and thederivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolicstructure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time dependent analyticregularity.
We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show its dense range in L^2 when acting on an appropriate subdomain.
We consider the Helmholtz equation $-Δu+V \, u - λ\, u = f $ on $\mathbb{R}^n$ where the potential $V:\mathbb{R}^n\to\mathbb{R}$ is constant on each of the half-spaces $\mathbb{R}^{n-1}\times (-\infty,0)$ and $\mathbb{R}^{n-1}\times (0,\infty)$. We prove an $L^p-L^q$-Limiting Absorption Principle for frequencies $λ>\max \, V$ with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations.
Spectral analysis of the discrete Maxwell operator: The limiting absorption principle
本研究针对三维正方晶格上的各向异性离散麦克斯韦算子开展谱分析，通过构造对应非零实值的共轭算子证明其满足极限吸收原理，并证实特定阈值下共轭算子本质自伴。
This paper is concerned with the time-dependent Maxwell's equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long …
We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions \begin{equation*} (L+λ)v=f, \qquad λ\in \mathbb{R} \end{equation*} under a Sommerfeld radiation condition at infinity. The operator $L$ is a second order elliptic operator with variable coefficients, the principal part is a small, long range perturbation of $-Δ$, while lower order terms can be singular and large. The main tool is a sharp uniform resolvent estimate, which has independent applications to the problem of embedded eigenvalues and to smoothing estima…
Limiting absorption principle and absence of eigenvalues for massless Klein-Gordon operators on perturbations of the Minkowski spacetime
本文在接近闵可夫斯基时空的弯曲时空中，利用Mourre交换子方法与Virial定理，证明了无质量Klein-Gordon算子的加权预解估计、极限预解式存在性、绝对连续性及特征值缺失。
We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y.Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.
In this paper, we study the mapping property form $L^p$ to $L^q$ of the resolvent of the Fourier multiplier operators and scattering theory of generalized Schrödinger operators. Though the first half of the subject is studied in [4], we extend their result to away from the duality line and we also study the Hölder continuity of the resolvent.
We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum', as well as the `off-spectrum', spectral family is Fredholm in function spaces which encode the Lagrangian regularity of generalizations of `outgoing spherical waves' of scattering theory, and indeed this persists in the `physical half plane'.
For the positive solutions of the competitive Gross-Pitaevskii system of two equations, we prove that L^\infty boundedness implies uniform Hölder boundedness as the competition parameter goes to infinity. Moreover we prove that the limiting profile is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with more than two densities are given.
Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains
针对拟柱状区域中的Dirichlet Laplacians，通过微分同胚变换和有限长完美匹配层，建立了极限吸收原理并证明了近似解指数收敛于出射/入射解。
We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dis- persion curves, that are real analytic functions. Each of them is simple and monotically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator both outside and at the thresholds. Finally, we establish analytic …
Several results about positive solutions -in a Lipschitz domain- of a nonlinear elliptic equation in a general form $ Δu(x)-g(x,u(x))=0$ are proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$, and a smooth domain. Our results include a characterization -in terms of a natural capacity- of a (conditional) removability property, a characterization of moderate solutions and of their boundary trace and a property relating arbitrary positive solutions to moderate solutions. The proofs combine techniques of non-linear p.d.e.\ with potential theoretic methods with respect to l…
Limiting absorption principle for time-harmonic elastic scattering of plane waves from diffraction gratings
该研究针对周期刚性衍射光栅的时谐弹性散射建立极限吸收原理，通过频率小正虚部扰动正则化不适定问题，得到共振频率下物理解的严格选择机制，框架可推广至多类边界条件。
Nonlocal Schrödinger-Poisson systems in $\mathbb R^N$: the fractional Sobolev limiting case
本文研究全空间中非局部 Schrödinger-Poisson 系统在分数阶 Sobolev 极限情形下的正解存在性，通过变分逼近方法处理符号变化对数 Riesz 核带来的困难，并建立了解的对称性、正则性与衰减性。
An Effective SPDE Model for a Schrödinger Equation with a Fluctuating Magnetic Potential
本文针对随机波动外磁场中带电粒子的薛定谔方程，提出一种结合确定性平均与波动行为的随机薛定谔方程，证明其与原模型极限行为一致，为研究相关可观测量分布提供了简便方法。
In this paper we prove dispersive estimates for the system formed by two coupled discrete Schrödinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals.
In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in \cite{Ng-Complementary}, we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. There approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equ…
From nonlinear Schrödinger equation to interacting particle system: 1 < p < 2
该论文研究非线性薛定谔方程在无穷多峰情形下的极限行为，推导峰点相互作用律，填补了 1 < p < 2 区间的空白，扩展了已有工作。
We consider the scattering theory for the Schrodinger equation with $-Δ-|x|^α$ as a reference Hamiltonian, for $0< α\leq 2$, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of $α$, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity an…
Existence and Limiting Profiles of Normalized Travelling Wave Solutions for the Pseudo-Relativistic Schrödinger Equation with Logarithmic Nonlinearity
研究具有对数非线性的伪相对论薛定谔方程归一化解，用变分方法对其极小值存在性分类，给出质量临界情形阈值，分析爆破行为及不同情形下极小值的极限轮廓。
Limiting Absorption Principle for the Helmholtz Equation with Sign-Changing Coefficients in Multilayer Spheres
针对多层球壳中折射率正负交替的亥姆霍兹问题，通过构造T-强制算子恢复强制性，并集成复波数Dirichlet-to-Neumann算子处理无界域紧性缺失，结合变分框架与先验估计严格建立极限吸收原理，证明传输问题的适定性，并通过最优迹常数分析唯一性对几何的依赖，为多层超材料设计提供数学准则。
Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger Equations
本文研究非齐次分数阶 Schrödinger 方程的 $L^2$ 约束极小化问题，刻画临界参数 $a^*$ 处极小元的存在性分类及非负极小元的极限行为。
Limiting absorption principles and linear inviscid damping in the Euler-Boussinesq system in the periodic channel
本文研究周期通道中Euler-Boussinesq系统在线性分层Couette流附近的线性化问题，利用极限吸收原理证明任意正Richardson数下扰动密度和速度场的最优无粘阻尼，并精确刻画了线性化算子的谱结构。
We analyze the thermodynamics of the focusing discrete nonlinear Schrödinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$ and under a model with two parameters, representing inverse temperature and strength of the nonlinearity, respectively. We prove the existence of limiting free energy and analyze the phase diagram for general $d,p$. We also prove the existence of a continuous phase transition curve that divides the parametric plane into two regions involving the appearance or non-appearance of solitons. Appropriate upper and lower bounds for the curve are constructed. We …
We construct solutions to the nonlinear magnetic Schrödinger equation $$ \left\{ \begin{aligned} - \varepsilon^2 Δ_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & &\text{in}\ Ω,\\ u &= 0 & &\text{on}\ \partialΩ, \end{aligned} \right. $$ in the semiclassical régime with strong magnetic fields. In contrast with the well-studied mild magnetic field régime, the limiting energy depends on the magnetic field allowing to recover the Lorentz force in the semi-classical limit. Our solutions concentrate around global or local minima of a limiting energy that depends on the electric potential and t…
Limiting absorption principle for time-harmonic acoustic and electromagnetic scattering of plane waves from a bi-periodic inhomogeneous layer
本文研究了双周期非均匀层中时谐声学和电磁散射问题，证明了极限吸收原理在存在束缚态情况下的有效性，以确保散射问题的唯一性。
Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation
本文研究了分数霍尔次方程的辐射条件，通过极限吸收原理计算了自由空间格林函数，并证明了其渐进行为与经典霍尔次方程一致，为分数阶霍尔次方程的散射理论提供了理论基础。
We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schrödinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \| R(λ+i\eps) f \|_{H^{0,-1/2-σ}} \leq C(λ, H) \| f \|_{H^{0,1/2+σ}}.$$ We are particularly interested in the exact nature of the dependence of the constants $C(λ,H)$ on both $λ$ and $H$. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies $λ\ll 1$, medium energies $λ\sim 1$, and large energies $λ\gg 1$,…
We study the Cauchy problem of the mass critical nonlinear Schrodinger equation with derivative with the 4pi mass. One has the global well-posedness in H^1 whenever "the mass is strictly less than 4pi" or whenever "the mass is equal to 4pi and the momentum is strictly less than zero". In this paper, by the concentration compact principle as originally done by Kenig-Merle, we obtain the limiting profile of blow up solutions with the critical 4pi mass.
We study the limiting behavior of the two-dimensional singular stochastic stochastic cubic nonlinear complex Ginzburg-Landau with Gibbs measure initial data. We show that in the appropriate small viscosity and small noise regimes, the limiting dynamics is given by the deterministic cubic nonlinear Schrödinger equation at Gibbs equilibrium. In order to obtain this convergence, our approach combines both heat and Schrödinger analysis, within the framework of the Fourier restriction norm method of Bourgain (1993).
This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i\partial_t u+Δu= θ(ωt)\big(e^{4π|u|^2}-1\big)$, where $θ$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 \in H^1(\mathbb{R}^2)$, the solution $u_ω$ converges, as $|ω|$ tends to infinity to the solution $U$ of the limiting equation $i\partial_t U+ΔU= I(θ)\big(e^{4π|U|^2}-1\big)$ with the same initial data, where $I(θ)$ is the average of $θ$.
Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction
该研究针对带任意形状收缩自聚焦核的薛定谔方程，分析其正解与结点解的存在性和集中性，通过尺度变换得到对应非自治半线性椭圆方程的极限轮廓，并刻画了最小能量极限轮廓的对称性与衰减特性。
A Uniform Pole-Subtracted Limiting Absorption Principle for High-Contrast Elastic Resonator Clusters
本文针对高对比度弹性亚波长谐振器簇，建立极点扣除极限吸收原理，分解截断预解式并分析共振特性，揭示暗-亮共振混合规律。
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