Hongjie Dong, Wolfgang Staubach
We obtain unique continuation results for Schrodinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear Schrodinger equations.
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In this paper, Hardy's uncertainty principle and unique continuation properties of abstract Schrödinger equations in vector-valued classes are obtained
This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable coefficient versions of fractional Schrödinger equations.
In this paper we study unique continuation theorems for magnetic Schrödinger equation via Carleman estimates. We use integration by parts techniques in order to show these estimates. We consider electric and magnetic potentials with strong singularities at the origin and some decay at infinity.
We prove unique continuation properties for solutions of the evolution Schrödinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi-linear Schrödinger equations.
Quantitative strong unique continuation for the Schrödinger operator with unbounded potential
本文通过重访JK定理6.3，建立了具有无界势的薛定谔算子的定量强唯一延拓，并结合全局定量唯一延拓得到全局定量强唯一延拓。
Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.
We prove the unique continuation property for the differential inequality $|(-Δ)^{α/2}u|\leq|V(x)u|$, where $0<α<n$ and $V\in L_{\textrm{loc}}^{n/α,\infty}(\mathbb{R}^n)$, $n\geq3$.
Unique continuation inequalities for the Dunkl-Schrödinger equation via uncertainty principles
本文基于Dunkl变换的定量不确定性原理，建立了Dunkl-薛定谔方程在两个时间点的唯一持续不等式，证明了(\varepsilon,k)-薄集对Dunkl变换形成强 annihilating 对，并由此获得Dunkl-薛定谔方程解的定量唯一持续性质。
Strong Unique Continuation for Fractional Schrödinger Operators
该研究证明了0<s<1时，具尺度临界位势的分数阶薛定谔方程的强唯一延拓性质，是经典薛定谔算子Jerison-Kenig结果的非局部对应。
In this note we study the property of unique continuation for solutions of $|(-Δ)^{α/2}u|\leq|Vu|$, where $V$ is in a function class of potentials including $\bigcup_{p>n/α}L^p(\mathbb{R}^n)$ for $n-1\leqα<n$. In particular, when $n=2$, our result gives a unique continuation theorem for the fractional ($1<α<2$) Schrödinger operator $(-Δ)^{α/2}+V(x)$ in the full range of $α$ values.
We obtain a unique continuation result for the differential inequality $| (i\partial_t +Δ)u | \leq |Vu| + | W\cdot\nabla u |$ by establishing $L^2$ Carleman estimates. Here, $V$ is a scalar function and $W$ is a vector function, which may be time-dependent or time-independent. As a consequence, we give a similar result for the magnetic Schrödinger equation.
We obtain a global unique continuation result for the differential inequality $|(i\partial_t+Δ)u|\leq|V(x)u|$ in $\mathbb{R}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials $V(x)$ in $\mathbb{R}^{n}$. Our method is based on a new type of Carleman estimates for the operator $i\partial_t+Δ$ on $\mathbb{R}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.
Quantitative uniqueness of continuation for the Schrödinger equation : explicit dependence on the potential
本文针对具有有界或无界势的薛定谔方程，建立了从内部数据唯一延拓的定量不等式，其常数显式依赖于势，并将结果推广到椭圆方程及柯西数据情形。
We prove a logarithmic convexity result for exponentially weighted $L^2$-norms of solutions to electromagnetic Schrödinger equation, without needing to assume smallness of the magnetic potential. As a consequence, we can prove a unique continuation result in the style of the Hardy uncertainty principle, which generalizes the analogous theorems which have been recently proved by Escauriaza, Kenig, Ponce and Vega.
We study uniqueness properties of solutions of Schrödinger equations. The aim is to obtain sufficient conditions on the decay behavior of the difference of two solution $u_1-u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee the uniqueness of the solution, i.e. that $u_1\equiv u_2$.
Observability and unique continuation inequalities for the Schrödinger equations with inverse-square potentials
该研究将自由 Schrödinger 方程的能观性与唯一延拓不等式推广到半轴上的反平方势情形，并利用解表示、Hankel 变换不确定性原理及紧支集插值不等式完成证明。
Unique Continuation Properties from one time for hyperbolic Schrödinger equations
针对含时变复电场和定常实磁场的双曲型 Schrödinger 方程，证明单一时刻有界区域内的正质量会向外传播，并在时空柱体适当平均下建立解的高斯下界。
We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $χ_{(-\infty,E]}(H_L)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-s…
We study two types of unique continuation properties for the higher order Schrödinger equation with potential $$ i\partial_tu=(-Δ_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$ has certain exponential decay at two times, then $u\equiv0$, and this result is sharp by constructing critical non-trivial solutions. The second one says if $u\equiv0$ in an arbitrary half-space of $\mathbb{R}^{1+n}$, then $u\equiv0$ identically. The uniqueness theorems are given when $n=1$, but we also prove partial results when $n\in\mathbb{N}_+$ for their own intere…
Unique continuation and Hardy's uncertainty principle for hyperbolic Schrödinger equations
本研究将经典薛定谔方程的Hardy不确定性原理相关唯一延拓结果拓展至双曲型薛定谔方程，证明满足双时刻高斯衰减的解恒为零，通过调整Carleman估计技术克服了双曲特性带来的证明难点。
Bifurcation and global continuation of travelling-rotating Schrödinger maps on the sphere
研究球面薛定谔映射方程的行波旋转解，利用两个首次积分将剖面方程化为标量三次方程，证明从赤道分支在特定参数处分岔，并建立全局延拓替代，数值延拓表明收敛到北极边界。
Unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition
本文证明在几何控制条件下，半线性薛定谔方程的解具有有限时间全局解析性传播，从而获得唯一延拓、半全局控制与镇定，并肯定回答了2006年的开放问题。
In this paper, Morgan type uncertainty principle and unique continuation properties of abstract Schrödinger equations with time dependent potentials in vector-valued classes are obtained. The equation involves a possible linear operators considered in the Hilbert spaces. So, by choosing the corresponding spaces H and operators we derived unique continuation properties for numerous classes of Schrödinger type equations and its systems which occur in a wide variety of physical systems
In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, $(M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary…
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for $Δ+ V$. That is, for any non-trivial $u$ that solves $Δu + V u = 0$ in some open, connected subset of $\mathbb{R}^n$, we estimate the vanishing order of solutions in terms of the $L^t$-norm of $V$. Our results apply to all $t > \frac n 2$ and $n \ge 3$. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique con…
We prove quantitative unique continuation results for solutions of $Δw - k^2 w = V w + W\cdot \nabla w$ in a neighborhood of infinity, where $k > 0$, and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim |x|^{-N}$ and $|W(x)| \lesssim |x|^{-P}$ for some $N, P > 1$. For $M(R, 4n/k) = \inf \{||w||_{L^2(B_{4n/k}(x_0))} : |x_0| = R \}$, we show that if the solution $w$ is non-zero, bounded, and normalized, then $M(R, 4n/k) \gtrsim \exp(-kR - G \log R)$, where $G > \frac{n-1}{2}$ is a constant. An examination of radial solutions to $Δw - k^2 w = V w + W\cdot \nabla w$…
In this paper, Hardy's uncertainty principle and unique continuation properties of Schrodinger equations with operator potentials in Hilbert space-valued classes are obtained. Since the Hilbert space H and linear operators are arbitrary, by choosing the appropriate spaces and operators we obtain numerous classes of Schrodinger type equations and its finite and infinite many systems which occur in a wide variety of physical systems.
In this article we prove the property of unique continuation (also known for C^\infty functions as quasianalyticity) for solutions of the differential inequality |Δu| \leq |Vu| for V from a wide class of potentials (including L^{d/2,\infty}_{\loc}(R^d) class) and u in a space of solutions Y_V containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?
We obtain a unique continuation result for fractional Schrödinger operators with potential in Morrey spaces. This is based on Carleman inequalities for fractional Laplacians.
We consider the nonlinear stationary Schrödinger equation \begin{equation*} -Δu -λu= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N \end{equation*} in the case where $N \geq 3$, $p$ is a superlinear, subcritical exponent, $Q$ is a bounded, nonnegative and nontrivial weight function with compact support in $\mathbb{R}^N$ and $λ\in \mathbb{R}$ is a parameter. Under further restrictions either on the exponent $p$ or on the shape of $Q$, we establish the existence of a continuous branch $\mathcal{C}$ of nontrivial solutions to this equation which intersects $\{λ\} \times L^{s}(\mathbb{R}^N)$ for ev…
We prove the strong unique continuation property for many-body Schrödinger operators with an external potential and an interaction potential both in $L^p_{\rm loc}(\mathbb{R}^d)$, where $p > \max(2d/3,2)$, independently of the number of particles. With the same assumptions, we obtain the Hohenberg-Kohn theorem, which is one of the most fundamental results in Density Functional Theory.
Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schrödinger operators and control theory. We review recent results and announce new ones regarding quantitative unique continuation principles for partial differential equations with an underlying multiscale structure. They concern Schrödinger and second order elliptic operators. An important feature is that the estimates are scale free and with quantitative dependence on parameters. These unique continuation principles apply to functions satisfying certain `r…
We prove quantitative unique continuation results for solutions of $-Δu + W\cdot \nabla u + Vu = λu$, where $λ\in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle x\rangle^{-N}$ and $|W(x)| \lesssim \langle x\rangle^{-P}$. For $M(R) = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, we show that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \gtrsim \exp(-C R^{β_0}(\log R)^{A( R)})$, where $β_0 = \max\{2 - 2P, \frac{4-2N}{3}, 1\}$. Under certain conditions on $N$, $P$ and $λ$, we construct examples (some of which are in the s…
We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the so-called strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived by using a linearization procedure.
We establish a strong unique continuation property for stochastic parabolic equations. Our method is based on a suitable stochastic version of Carleman estimate. As far as we know, this is the first result for strong unique continuation property of stochastic partial differential equations.
Consider the higher order parabolic operator $\partial_t+(-Δ_x)^m$ and the higher order Schrödinger operator $i^{-1}\partial_t+(-Δ_x)^m$ in $X=\{(t,x)\in\mathbb{R}^{1+n};~|t|<A,|x_n|<B\}$, where $m$ and $n$ are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when $x_n>0$, then the solutions vanish in $X$. Such results are global if $n>1$, and we also prove some relevant local results.
In this paper, we derive a local unique continuation property for stochastic hyperbolic equations without boundary conditions. This result is proved by a global Carleman estimate.
Unique continuation at infinity for Schrödinger equations with Reverse Hölder Potentials
本文研究具有逆Hölder势的薛定谔方程，证明若解在无穷远处增长不超过由Agmon距离确定的阈值，则解必平凡，该结果属于Liouville型定理，与Landis猜想相关。
In this paper we establish some new $L^{2}-L^{2}$ Carleman estimates for the Baouendi-Grushin operators $\mathscr{B}_γ$, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique continuation; (ii) the strong unique continuation property for some degenerate sublinear equations.
We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.
This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order $α\in(0,1)$. Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.
This work is devoted to the strong unique continuation problem for second order parabolic equations with nonsmooth coefficients. Introduction and bibliography have been revised.
We consider a time-dependent structured population model equation and establish a Carleman estimate. We apply the Carleman estimate to prove the unique continuation which means that Cauchy data on any lateral boundary determine the solution uniquely.
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems.
Comments on the quantitative uniqueness of continuation for evolution equations
本文通过结合可观测性不等式与能量估计，为侧边界消失的演化方程解建立了接近最优的延拓定量唯一性结果。
We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing monotonicity of some Almgren-type frequency functional via an extension procedure.
We prove that solutions to elliptic equations in two variables in divergence form, possibly non-selfadjoint and with lower order terms, satisfy the strong unique continuation property.
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong unique continuation property. Moreover, we also discuss the unique continuation property from measurable sets, which shows that nodal domains to these equations must have vanishing Lebesgue measure. Our methods rely on suitable Carleman estimates, for which we include the sublinear potential into the main part of the operator.
Unique continuation for a mean field game system
该研究针对平均场博弈系统，证明了其唯一延拓性，即任意侧向子边界上柯西数据为零时解恒为零，丰富了相关数学理论。
Unique continuation for the momentum ray transform
本文利用分数阶拉普拉斯算子的唯一延拓性质，证明了动量射线变换作用于张量的保守性以及加权射线与锥变换的反局部性。
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.
In this survey we discuss the frequency function method so as to study the problem of unique continuation for elliptic partial differential equations. The methods used in the note were mainly introduced by Garofalo and Lin.
In this paper, we prove the unique continuation property for the weak solution of the plate equation with non-smooth coefficients. Then, we apply this result to study the global attractor for the semilinear plate equation with a localized damping.
In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.
A logarithmic convexity approach to quantitative unique continuation for the complex Ginzburg-Landau operator
研究复金兹堡 - 朗道方程解的定量唯一延拓估计，用对数凸性方法证明该性质，导出明确常数并刻画其参数依赖性。
We prove unique continuation principles for solutions of evolution Schrödinger equations with time dependent potentials. These correspond to uncertainly principles of Paley-Wiener type for the Fourier transform. Our results extends to a large class of semi-linear Schrödinger equation.
Determining Schrodinger Potentials on Riemannian Manifolds
本文研究从Dirichlet-to-Neumann映射确定黎曼流形上薛定谔方程势的逆问题，提出两个唯一可辨识性结果，分别基于相对傅里叶条件和乘积型可加分离假设，推动该开放问题进展。
New quantitative unique continuation result for elliptic equations
本文基于Carleman不等式，通过简单直接的证明方法，为椭圆方程提供了新的定量化唯一持续性结果，并展示了斯托克斯方程的类似结果。
We prove the following unique continuation result: if a solution to the level set equation for mean curvature flow in a mean-convex domain agrees to infinite order at the point where it attains its maximum with the solution for a ball, then it agrees everywhere and the domain is a ball.
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