Thomas Duyckaerts
We construct a potential $V$ on $\RR^d$, smooth away from one pole, and a sequence of quasi-modes for the operator $-Δ+V$, which concentrate on this pole. No smoothing effect, Strichartz estimates nor dispersive inequalities hold for the corresponding Schrodinger equation.
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Let P be the operator $-Δ+V$ on R^d, where $V$ is a real potential with several inverse square singularities. The usual non-trapping type high-frequency inequality on the truncated resolvent of $P$ is shown, using semi-classical measures. This inequality implies local smoothing effect on the corresponding Schrodinger equation.
In this paper, we consider Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4,using singular Adams inequality and variational techniques, we get the existence of ground state solutions. Moreover, we also get the same results without the Ambrosetti-Rabinowitz (AR) condition.
This paper is devoted to the study of essential self-adjointness of a relativistic Schrödinger operator with a singular homogeneous potential. From an explicit condition on the coefficient of the singular term, we provide a sufficient and necessary condition for essential self-adjointness.
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.
In this paper, we consider an optimal bilinear control problem for the nonlinear Schrödinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our results generalize the ones in \cite{Sp} in several aspects.
We investigate existence and qualitative behaviour of solutions to nonlinear Schrödinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of degree -1, including the Aharonov-Bohm class. In particular, by variational arguments we prove a result of multiplicity of solutions distinguished by symmetry properties.
In this paper we consider the space-fractional Schrödinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper is a continuation and extension of a previous one where the classical Schrödinger equation on $\mathbb R^n$ with singular potentials was considered.
In this paper, the space-fractional Schrödinger equations with singular potentials are studied. Delta-like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of 'weakened' solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particle accumulating effect is observed in the singular cases. From the mathematical point of view, a "splitting of the strong singularity" phenomena is also observed.
We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic NLS. These solutions have a quartic-root blowup rate, and collapse with a quasi self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.
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 study the spectrum of the one-dimensional Schrödinger operator $H_0$ with a matrix singular distributional potential $q=Q'$ where $Q\in L^{2}_{\mathrm{loc}}(\mathbb{R},\mathbb{C}^{m})$. We obtain generalizations of Ismagilov's localization principles, which give necessary and sufficient conditions for the spectrum of $H_0$ to be bounded below and discrete.
We present new singular solutions of the biharmonic nonlinear Schrodinger equation in dimension d and nonlinearity exponent 2σ+1. These solutions collapse with the quasi self-similar ring profile, with ring width L(t) that vanishes at singularity, and radius proportional to L^α, where α=(4-σ)/(σ(d-1)). The blowup rate of these solutions is 1/(3+α) for 4/d\leσ<4, and slightly faster than 1/4 for σ=4. These solutions are analogous to the ring-type solutions of the nonlinear Schrodinger equation.
Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.
We are interested in nonlinear fractional Schrödinger equations with singular potential of form \begin{equation*} (-Δ)^su=\fracλ{|x|^α}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$, $α>0$, $p\ge1$ and $λ\in \mathbf R$. Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $α$, $p$ and $λ$.
We study the Cauchy problem for the non-linear Schrödinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific interest is give to the point-like $δ$ and $δ'$ impurity and for two $δ$-interactions in one dimension. We also consider the periodic case which is analyzed in a functional space based on Fourier transform and local-in-time well-posedness is proved.
In this paper we prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth \begin{eqnarray*} -Δu-λc(x)u-κα(Δ(|u|^{2α}))|u|^{2α-2}u = |u|^{q-2}u+|u|^{2^*-2}u,\quad u\in{D^{1,2}(\mathbb{R}}^N), \end{eqnarray*} via variational methods, where $λ\geq0$, $0<α<1/2$, $2<q<2^*$. It is interesting that we do not need to add a weight function to control $|u|^{q-2}u$.
We study the local behavior of solutions of the stationary Schr\" od\-inger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this result with a quantitative unique continuation principle for singular potentials we obtain log-Hölder continuity for the density of states outer-measure in one, two, and three dimensions for Schr\" odinger operators with singular potentials, results that hold for the density of states measure when it exists.
There have been a lot of works concerning the Strichartz estimates for the perturbed Schrödinger equation by potential. These can be basically carried out adopting the well-known procedure for obtaining the Strichartz estimates from the weighted $L^2$ resolvent estimates for the Laplacian. In this paper we handle the Strichartz estimates without relying on the resolvent estimates. This enables us to consider various potential classes such as the Morrey-Campanato classes.
We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. Since the unified approach is known as providing efficient solutions to…
In this work we investigate positivity properties of nonlocal Schrödinger type operators, driven by the fractional Laplacian, with multipolar, critical, and locally homogeneous potentials. On one hand, we develop a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, while, on the other hand, we study the localization of binding for this kind of potentials. Combining these two tools and performing an inductive procedure on the number of poles, we establish necessary and sufficient conditions for the existence of a configur…
We consider the semilinear electromagnetic Schrödinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(Ω,\mathbb{C}), where $Ω=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq m\leq N$, $N\geq3$, $2^{\ast}:= 2N/(N-2)$ is the critical Sobolev exponent, $V$ is a Hardy term and $A$ is a singular magnetic potential of a particular form which includes the Aharonov-Bohm potentials. Under some symmetry assumptions on $A$ we obtain multiplicity of solutions satisfying certain symmetry properties.
The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schrödinger-Poisson system {ll} -Δu +(λ+\frac{1}{|y|^α})u+ϕ(x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -Δϕ= u^2,\ \lim\limits_{|x|\rightarrow +\infty}ϕ(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $λ\geqslant0$, $α\in[0,8)$ and $\max\{2,\frac{2+α}{2}\}<p<5$.
The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, point-wise upper estimate are derived.
Schrödinger equation for Sturm-Liouville operator with singular propagation and potential
本文研究带奇异传播子与势的分数阶Sturm-Liouville算子对应的Schrödinger方程初边值问题，正则情形下用分离变量法结合Fourier级数构造解并得Sobolev估计，强奇异情形下引入极弱解概念并证明其存在唯一等性质。
Weighted inequalities for Schrödinger type Singular Integrals on variable Lebesgue spaces
本文研究由薛定谔算子生成的奇异积分算子在加权变指数勒贝格空间中的有界性，位势属于逆赫尔德类，核满足与临界半径函数相关的尺寸和光滑性条件。
We consider the following nonlinear Schrödinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2σ}}\log|x|u=0 \] in $\mathbb{R}^N$. From the classical argument, the solution with subcritical mass ($\|u\|_2<\|Q\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$. Here, $Q$ is the ground state of the mass-critical problem. Therefore, we are interested in the existence and behaviour of blow-up solutions for the threshold ($\left\|u_0\right\|_2=\left\|Q\right\|_2$).
This paper deals with global dispersive properties of Schrödinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey-Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the …
In this short note, we prove Strichartz estimates for Schrödinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than two. The main ingredient of the proof is a variant of Kato's smoothing estimate with a singular weight.
Motivated mainly by the localization over an open bounded set $Ω$ of $\mathbb R^n$ of solutions of the Schrödinger equations, we consider the Schrödinger equation over $Ω$ with a very singular potential $V(x) \ge C d (x, \partial Ω)^{-r}$ with $r\ge 2$ and a convective flow $\vec U$. We prove the existence and uniqueness of a very weak solution of the equation, when the right hand side datum $f(x)$ is in $L^1 (Ω, d(\cdot, \partial Ω))$, even if no boundary condition is a priori prescribed. We prove that, in fact, the solution necessarily satisfies (in a suitable way) the Dirichlet condition $u…
We prove the existence of ground state solutions for the nonlinear Schrodinger-Maxwell equations with a singular potential.
The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity
针对强奇性三维能量临界 INLS 的未解 Cauchy 问题，论文改进加权空间上的非齐次 Strichartz 估计，建立了 $3/2\leq α<11/6$ 范围内的局部及小数据全局适定性。
The aim of this paper is to provide uniform estimates for the eigenvalue spacings of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.
We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation $$ \begin{array}{c} -Δu +V(x)u=λa(x)u^{-γ}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array} $$ beyond the Nehari extremal value, as defined in Il'yasov [Topol. Methods Nonlinear Anal., 2017], when the potential $b \in L^{\infty}(\mathbb{R}^{N})$ may change its sign, $0<a\in L^{\frac{2}{1+γ}}(\mathbb{R}^{N})$, $V$ is a positive continuous function, $N\geq 3$ and $λ>0$ is a real parameter. The main difficulties come from the non-differentiability of the energy functional…
We show that the subcritical $d$-dimensional nonlinear Schrödinger equation $i ψ_t + Δψ+ |ψ|^{2 σ} ψ= 0$, where $1<σd<2$, admits smooth solutions that become singular in~$L^p$ for $p^*<p \le \infty$, where $p^*:=\frac{σd}{σd -1}$. Since $\lim_{σd \to 2-} p^* = 2$, these solutions can collapse at any $2<p \le \infty$, and in particular for $p = 2 σ+2$.
Consider operators of the form $L^{γV}:=Δ+γV$ in a bounded Lipschitz domain $Ω\subset \mathbb{R}^N$. Assume that $V\in C^1(Ω)$ satisfies $|V(x)| \leq \bar a \,\mathrm{distance}\,(x,\partialΩ)^{-2}$ for every $x\in Ω$ and $γ$ is a number in a range $(γ_-,γ_+)$ described in the introduction. The model case is $V(x)= \mathrm{distance}\,(x,F)^{-2}$ where $F$ is a closed subset of $\partialΩ$ and $γ< c_H(V)=$ Hardy constant for $V$. We provide sharp two sided estimates of the Green and Martin kernel for $L^{γV}$ in $Ω$. In addition we establish a pointwise version of the 3G inequality.
In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=Δ^2-V=Δ^2-c|x|^{-4}$ where $c$ is any constant such that $c<\left(\frac{N(N-4)}{4}\right)^2$. The semigroup generated by $-A$ in $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $ΔA^{-1/2}$ on $L^p(\mathbb{R}^N)$ for all $p\in(p_0^{'},2]$.
Asymptotics of solutions to Schroedinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.
Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $K \subset Ω$ be a compact, $C^2$ submanifold in $\mathbb{R}^N$ without boundary, of dimension $k$ with $0\leq k < N-2$. We consider the Schrödinger operator $L_μ= Δ+ μd_K^{-2}$ in $Ω\setminus K$, where $d_K(x) = \text{dist}(x,K)$. The optimal Hardy constant $H=(N-k-2)/2$ is deeply involved in the study of $-L_μ$. When $μ\leq H^2$, we establish sharp, two-sided estimates for Green kernel and Martin kernel of $-L_μ$. We use these estimates to prove the existence, uniqueness and a priori estimates of the solution to the bound…
Consider operators $L^{V}:=Δ+ V$ in a bounded Lipschitz domain $Ω\subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(Ω)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partialΩ)^{-2}$ in $Ω$ and a second condition that guarantees the existence of a ground state $Φ_V$. If, for example, $V>0$ this condition reads $1<c_H(V)$ (= the Hardy constant relative to $V$). We derive estimates of positive $L_V$ harmonic functions and of positive Green potentials of measures $τ\in {\mathfrak M}_+(Ω;Φ_V)$. These imply estimates of positive $L_V$ supersolutions and of $L_V$ subsolutions. Similar resu…
We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile…
In this paper we study the existence of solutions and their concentration phenomena of a singularly perturbed semilinear Schrodinger equation with the presence of the critical Sobolev exponent.
Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schrödinger equation
本文通过凸积分构造了一维周期三次非线性薛定谔方程的奇异解，证明其在 $H^{1/6}(\mathbb T)$ 阈值下的强非唯一性，补充了该方程奇异解的相关理论结果。
This paper deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. When the poles form a symmetric structure, it is natural we wonder how the symmetry affects such mutual interaction. The present paper means to study this aspect from the point of view of the existence of solutions inheriting the same symmetry properties as the set of singularities.
Boundedness of wave operators for Schrödinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.
Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
本文针对具有奇异复势的薛定谔方程，通过建立尺度不变的 Carleman 不等式证明二倍不等式，并结合多尺度论证实现小性传播，进而给出特征函数对数的 BMO 范数上界。
We consider the steady fractional Schrödinger equation $L u + V u = f$ posed on a bounded domain $Ω$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-Δ)^s$; $V\ge 0$ is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of $(-Δ)^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-Δ)^s u + V u = μ$ exists for every Borel measure $μ$. On the other hand, when $V$ is allowed to be more …
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator $\mathcal{L}=-Δ+ \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the…
Numerical computation of solutions of the critical nonlinear Schrodinger equation after the singularity
采用Mori-Zwanzig形式构建约化模型，数值计算一维临界非线性薛定谔方程在有限时间奇点后的解，结果与Tao构造的后奇点解特征一致。
In this paper, we study spectral properties of a family of quasi-periodic Schrodinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular.
We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.
The paper discusses nonlinear singular perturbations of delta type of the fractional Schrödinger equation $\imath\partial_tψ=\left(-\triangle\right)^sψ$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.
Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
本文将薛定谔算子谱子空间的定量唯一延拓估计推广到含奇异势的情形，利用调整后的Carleman估计证明，并应用于随机算子的Wegner估计及奇异热源控制理论。
Propagation for Schrödinger operators with potentials singular along a hypersurface
研究Schrödinger算子沿超曲面具有共法向奇异势时的缺陷测度传播，证明横穿或一阶相切情形下标准传播定理在势函数绝对连续条件下成立。
Uncertainty Principles, Spectral Localization, and Singular Schrödinger Operators on Compact Manifolds
通过结合正交系统的限制估计与拉普拉斯-贝尔特拉米和薛定谔算子的谱投影界，建立了紧无边黎曼流形上的不确定性原理，并应用于不完全谱数据恢复的唯一性结果。
We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-Δu + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of Bénilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\{u = 0\}$ and on the existence o…
We generalize the Strichartz estimates for Schrödinger operators on compact manifolds of Burq, Gérard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times M)$-Strichartz estimates hold for $e^{-itH_V}$ when $H_V=-Δ_g+V(x)$ with $V\in L^{n/2}(M)$ if $n\ge3$ or $V\in L^{1+δ}(M)$, $δ>0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao theorem and $I\subset {\mathbb R}$ a bounded interval. We do this by formulating and proving new "quasimode" estimates for scaled dyadic unperturbed Schrödinger operators and taking advant…
We consider the Schrödinger operators $H_V=-Δ_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp error term $O(λ^{n-1})$ holds for potentials in $L^n(M)$.
Uniqueness for the Schrödinger equation with an inverse square potential and application to controllability and inverse problems
本文利用将 Schrödinger 方程转化为椭圆方程的技术，在无观测区域几何限制下证明反平方势奇异方程的精确唯一性，并据此建立近似可控性与反源问题唯一性。
We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ of $\varepsilon\mathbb{Z}^n$-periodic Schrödinger operators with $δ'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}^\varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions s…
We established a generalized version of the Christ-Kiselev's multi-linear operator technique to deal with the spectral theory of Schrödinger operators. As applications, several spectral results of perturbed periodic Schrödinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of the Hausdorff dimension of singular spectra.
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