Piero D'Ancona, Luca Fanelli
We prove a local in time smoothing estimate for a magnetic Schrodinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two derivatives. The proof is based on the multiplier method and no pseudofferential techniques are required.
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In this paper a global smoothing property of Schrodinger equations is established in the critical case in dimensions two and higher. It is shown that the critical smoothing estimate is attained if the smoothing operator has some structure. This structure is related to the geometric properties of the equations. Results for critical cases for operators of higher orders as well as for hyperbolic equations are also given.
In this paper we prove the smoothing effect for solutions of Schr{ö}dinger equations with variable coefficients and in a non trapping exterior domain. We allow quadratic potentials at infinity.
Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of [CM14] on inflection-transmission type trapping on warped product manifolds to the case of multi-warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has…
We prove $L^p$ and smoothing estimates for the resolvent of magnetic Schrödinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrödinger and Pauli operators with complex electromagnetic potentials.
We prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one.
We prove that the geometric control condition is not necessary to obtain the smoothing effect and the uniform stabilization for the strongly dissipative Schrödinger equation.
We prove, under the exterior geometric control condition, the Kato smoothing effect for solutions of an inhomogenous and damped Schrödinger equation on exterior domains.
Given a Schrödinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary conditions, respectively. Moreover, we prove a strict inequality between the eigenvalues of two different elliptic differential operators on the same domain with Dirichlet boundary conditions.
The aim of this note is to extend recent results of Yajima-Zhang \cite{Y-Z1, Y-Z2} on the 1/2- smoothing effect for Schrödinger equation with potential growing at infinity faster than quadratically.
We improve local smoothing estimates for fractional Schrödinger equations for $α\in (0,1) \cup (1,\infty)$.
We discuss the microlocal Gevrey smoothing effect for the Schrödinger equation with variable coefficients via the propagation property of the wave front set of homogenous type. We apply the microlocal exponential estimates in a Gevrey case to prove our result.
We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz bounds on the function parametrizing locally a chosen edge of the essential spectrum of a Schrödinger operator in dependence of a coupling constant. Analogous estimates for eigenvalues, possibly in…
We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schrödinger equation with constant coefficients which appears as a model for propagation of pulse in optical fiber.
The purpose of this paper is to study the effect of conformal perturbations on the local smoothing effect for the Schrödinger equation on surfaces of revolution. The paper \cite{ChWu-lsm} studied the Schrödinger equation on surfaces of revolution with one trapped orbit. The dynamics near this trapping were unstable, but degenerately so. Beginning from the metric $g$ from this paper, we consider the perturbed metric $g_s = e^{sf}g$, where $f$ is a smooth, compactly supported function. If $s$ is small enough and finitely many derivatives of $f$ satisfy appropriate symbolic estimates, then we sho…
We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial vel…
We show some new local smoothing estimates of the fractional Schrödinger equations with initial data in $α$-modulation spaces via decoupling inequalities. Furthermore, our necessary conditions show that the local smoothing estimates are sharp in some cases. As applications, the local smoothing estimates could show some new local well-posedness on modulation spaces of the fourth-order nonlinear Schrödinger equations on the line.
In this work we consider an example of a linear time-degenerate Schrödinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problems involving this operator and polynomial derivative nonlinearities are locally well-posed and their solutions also satisfy the same smoothing estimates as the linear solutions.
We consider the Cauchy problem of a class of higher order Schrödinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the persistence of regularity. We classify this class of equations into three types on the basis of their smoothing property.
Improved local smoothing estimate for the fractional Schrödinger operator
本文利用Guth的k-broad范数估计，改进了分数阶薛定谔算子（α>1）的局部光滑估计结果。
By means of a perturbation method recently introduced by Bolle, we discuss the existence of infinitely many solutions for a class of perturbed symmetric higher order Schrodinger equations with non-homogeneous boundary data on unbounded domains.
The Kato-Yajima smoothing estimate is a smoothing weighted $L^2$ estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey-Campanato weights. In this paper we make this generalization more sharp in terms of the so-called Kerman-Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted $L^2$ space. Obtained results are also extended to the fractional Schrödinger propagator.
The present paper is devoted to weighted Nonlinear Schrödinger- Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.
We consider some scale invariant generalizations of the smoothing estimates for the free Schrödnger equation obtained by Kenig, Ponce and Vega. Applying these estimates and using appropriate commutator estimates, we obtain similar scale invariant smoothing estimates for perturbed Schrödnger equation with small magnetic potential.
A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb T)$ solutions to the derivative nonlinear Schrödinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.
We consider the periodic non-linear Schrödinger equation with non-linearity given by $|u|^{p-1}u$ for odd $p > 1$ in dimension $1$. We first establish that the difference between the non-linear evolution and a phase rotation of the the linear evolution is in a smoother space. We then study forced and damped defocusing non-linear Schrödinger equations of the above type and establish an analogous smoothing statement that extends globally in time. As a corollary we establish both existence and smootheness for global attractors in the energy space.
In this note we consider the self-adjoint Schrödinger operator $\mathsf{A}_α$ in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $δ$-potential supported on a Lipschitz hypersurface $Σ\subseteq\mathbb{R}^d$ of strength $α\in L^p(Σ)+L^\infty(Σ)$. We show the uniqueness of the ground state and, under some additional conditions on the coefficient $α$ and the hypersurface $Σ$, we determine the essential spectrum of $\mathsf{A}_α$. In the special case that $Σ$ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schrödinger operator as Birman-Schwinger operator. As an application we pro…
In this article we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first reducibility results for an unbounded perturbation of a linear system which is not integrable.
A smoothing effect for the fractional Schrödinger equations on the circle and observability
本文研究圆周上分数阶 Schrödinger 方程在任意负指标 Sobolev 初值下的重整化解，通过光滑效应定义其模平方，并据此建立粗糙控制的能观性估计。
The purpose of this note is to prove global-in-time smoothing effects for the Schrödinger equation with potentials exhibiting critical singularity. A typical example of admissible potentials is the inverse-square potential $a|x|^{-2}$ with $a>-(n-2)^2/4$. This particularly gives an affirmative answer to a question raised by Bui-D'Ancona-Duong-Li-Ly [4]. The proof employs a uniform resolvent estimate proved by Barceló-Vega-Zubeldia [1] and an abstract perturbation method by Bouclet-Mizutani [3].
In this expository note, we prove some extensions and refinements of classical Kato type estimates with elementary techniques.
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schrödinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.
Determination of separable perturbations of an unbounded potential in the two-dimensional Schrödinger equation
利用狄利克雷-诺伊曼映射和卡尔曼不等式，证明了从边界数据唯一确定二维薛定谔方程中无界势的结果，并给出了定量版本。
Let $\mathcal{L}_a$ be a Schrödinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.
We prove smoothing estimates for Schrödinger equations $i\partial_t ϕ+\partial_x (a(x) \partial_x ϕ) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.
We show new local $L^p$-smoothing estimates for the Schrödinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and $L^p$-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in $L^p(\mathbb{R}^d)$ for $d \geq 1$ and $p > 2$, which was previously known for $d \geq 2$. The estimates are applied to …
We consider the control problem of the heat equation on bounded and unbounded domains, and more generally the corresponding inhomogeneous equation for the Schrödinger semigroup. We show that if the sequence of null-controls associated to an exhaustion of an unbounded domain converges, then the solutions do in the same way, and that the control cost estimate carries over to the limiting problem on the unbounded domain. This allows to infer the controllability on unbounded domains by studying the control problem on a sequence of bounded domains.
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。
We study the energy cascade problematic for some nonlinear Schrödinger equations on the torus $\T^2$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one $|u|^2u$. Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the f…
We consider the operator $H:= i \partial_t + \nabla \cdot (c \nabla)$ in an unbounded strip $Ω$ in $\mathbb{R}^2$, where $c(x,y) \in \mathcal{C}^3(\barΩ)$. We prove adapted a global Carleman estimate and an energy estimate for this operator. Using these estimates, we give a stability result for the diffusion coefficient $c(x,y)$.
Growth of Sobolev norms for unbounded perturbations of the Schrödinger equation on flat tori
该研究针对平坦环面上拉普拉斯算子的无界时变扰动问题，证明其Sobolev范数增长满足$\u27e8 t\u27e9^\u03b5$界，为相关偏微分方程解的正则性分析提供了理论结果。
Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials
本文研究带一般有界衰减势的高阶椭圆算子对应的薛定谔型方程，建立了其解的全局Kato光滑性估计与全套Strichartz估计，还引入了带附加导数项的新型Kenig-Ruiz-Sogge型一致Sobolev估计，结果可应用于非线性高阶薛定谔方程的研究。
Smoothing and Strichartz estimates for degenerate Schrödinger-type equations
本文研究时间退化Schrödinger型算子的基本估计，通过比较原理推导任意阶算子的整体齐次平滑估计，证明加权Strichartz型估计并应用于半线性Cauchy问题的局部适定性，同时涵盖非退化算子恢复标准结果。
We are interested in this article in investigating the smoothing effect properties of the solutions of the Schrodinger equation. We deduce global well-posedness results for the cubic Schrodinger equation in the exterior of several convex obstacles of R^{3}.
In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schrödinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are already known, but those of the local Kato smoothing estimates are not. Recently, Sun, Trélat, Zhang and Zhong have shown it only in spacetime $\mathbb R \times \mathbb R$. Our result resolves this …
We prove that the weakly damped cubic Schrödinger flow in $L^2(\T)$ provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak $ L^2(\T) $-convergence inspired by a previous work of the author. Combining the compactness in $ L^2(\T) $ of the attractor with the approach developed by Goubet, we show that the attractor is actually a compact set of $ H^2(\T) $. This asymptotic smoothing effect is optimal in view of the regularity of the steady states.
We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1<p<\infty$, of the Schrödinger type operator $A=(1+|x|^α)Δ-|x|^β$ with domain $D(A_p)=\{u\in W^{2,p}(\mathbb{R}^N): Au\in L^p(\mathbb{R}^N)\}$ generates a strongly continuous analytic semigroup provided that $N>2,\,α>2$ and $β>α-2$. Moreover this semigroup is consistent, irreducible, immediately compact and ultracontractive.
We consider refinements of the local smoothing estimates for the Schrödinger equation in domains which are exterior to a strictly convex obstacle in $\RR^n$. By restricting the solution to small, frequency dependent collars of the boundary, it is expected that taking its square integral in space-time should exhibit a larger gain in regularity when compared to the usual gain of half a derivative. By a result of Ivanovici, these refined local smoothing estimates are satisfied by solutions in the exterior of a ball. We show that when such estimates are valid, they can be combined with wave packet…
We study positive bound states for the semiclassical stationary nonlinear Schrödinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, the potentials might be singular at the origin or vanish superquadratically at infinity.
In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. In the aforementioned paper, the third author proved the local smoothing estimates for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compac…
We study the existence and concentration behavior of the bound states for the following logarithmic Schrödinger equation \begin{equation*} \begin{cases} -\varepsilon^2Δv+V(x)v=v\log v^2 \ \ &\text {in}\ \ \mathbb R^N,\\ v(x)\to 0 \ \ &\text {as}\ \ |x|\to\infty, \end{cases} \end{equation*} where $N\geq 1$, $\varepsilon>0$ is a small parameter, and $V$ may be unbounded below at infinity with a speed of at most quadratic strength. We show that around various types of local topological critical points of the potential function, positive bound state solutions exist and concentrate as $\varepsilon\…
In this paper we study the cubic fractional nonlinear Schrodinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a full derivative, while on the real line we get a derivative smoothing with an $ε$ loss. Our result lowers the regularity requirement of a recent theorem of Kappeler et al. on the periodic defocusing cubic N…
Strichartz estimates for Schrödinger equations with the multipole Aharonov--Bohm Hamiltonian
本文证明了$\mathbb{R}^2$上多极Aharonov-Bohm哈密顿量的Schrödinger方程的全局时间Strichartz估计，并通过建立局部光滑估计作为中间步骤。
In this paper we prove local-in-time Strichartz estimates with loss of derivatives for Schrödinger equations with variable coefficients and potentials, under the conditions that the geodesic flow is nontrapping and potentials grow polynomially at infinity. This is a generalization to the case with variable coefficients and improvement of the result by Yajima-Zhang. The proof is based on microlocal techniques including the semiclassical parametrix for a time scale depending on a spatial localization and the Littlewood-Paley type decomposition with respect to both of space and frequency.
In this paper we consider magnetic Schrödinger operators in R^n, n \ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not a corollary of the free case as the differentiated free resolvent does not have small operator norm on any weighted L^2 spaces. We instead show that the spectral radius of such operators decreases to zero, hence their perturbation of the identity is still invertible. The key estimates are based on an a…
We study the derivation of the Gibbs measure for the nonlinear Schrödinger equation (NLS) from many-body quantum thermal states in the high-temperature limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\mathbb{T}^d$ for $d=1,2,3$. This extends the author's earlier joint work with Fröhlich, Knowles, and Schlein, where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin, Nam, and Rougerie. Our proof is based on a perturbative ex…
The present paper is concerned with Schrödinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (resp. magnetic) potential can grow subquadratically (resp. sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. F…
This paper applies Hermite function techniques to give elementary proofs of Kato type smoothing estimates for the Schrödinger equation with quadratic potential in R^n+1. This is equivalent to proving a uniform L^2(R^n) to L^2(R^n) boundedness for a family of singularized Hermite projection kernels. As an applicationas the above estimate, we also prove the R^9 collapsing variable type Strichartz estimate.
We prove a sharp resolvent estimate in scale invariant norms of Amgon--Hörmander type for a magnetic Schrödinger operator on $\mathbb{R}^{n}$, $n\ge3$\begin{equation*} L=-(\partial+iA)^{2}+V \end{equation*}with large potentials $A,V$ of almost critical decay and regularity. The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schrödinger, wave and Klein--Gordon flows associated to $L$.
On the Well-posedness of Magnetic Schrödinger Equations with Unbounded Potentials
本文研究了在R^d上具有亚线性磁势和亚二次电势的磁Schrödinger方程的全局适定性，通过磁哈密顿流在相空间中近似解的方法，得出了在磁调和空间中的新结果。
This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrödinger equation.
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