Addolorata Salvatore, Marco Squassina
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.
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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.
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。
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 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.
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 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.
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…
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。
Biharmonic non-linear Schrödinger equation with an unbounded inhomogeneous term
本文研究了具有无界增长不均匀项的双调和非线性薛定谔方程，推导了非均匀Gagliardo-Nirenberg不等式，并揭示了解的全局存在与有限时间爆炸的二元性。
We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schrödinger map with values on the 2-D sphere, and to the 1-D cubic Schrödinger equation. Although these equations are completely integrable we show the existence of an unbounded growth of the energy density. The density is given by the amplitude of the high frequencies of the derivative of the tangent vectors of the curves, thus giving information of the oscillation at small scales. In the se…
We consider the following wave guide nonlinear Schrödinger equation, \begin{equation} (i\partial \_t+\partial \_{xx}-\vert D\_y\vert )U=\vert U\vert ^2U\ \tag{WS} \end{equation} on the spatial cylinder $\mathbb{R} \_x\times \mathbb{T} \_y$. We establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szegő equation. The proof is an adaptation of the method of Hani--Pausader--Tzvetkov--Visciglia. Combining this scattering theory with a recent result by Gérard--Grellier, we infer existence of global solutions to (WS) which are unbounded in t…
In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the fourth order Schrödinger type operator with unbounded coefficients of the form $$A=a^{2} Δ^2+V^{2}$$ where $a(x)=1+|x|^α$ and $V=|x|^β$ with $α>0$ and $β>(α-2)^+$. We obtain that $(-A,D(A))$ generates an analytic strongly continuous semigroup in $L^2(\mathbb{R}^N)$ for $N\geq5$. Moreover, the maximal domain $D(A)$ can be characterized for $N>8$ by the weighted Sobolev space \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N), |x|^{2α-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]
We summarize in these notes the course given at the Summer School of AIP 2019 held in Grenoble from July 1st to July 5th. This course was mainly devoted to the determination of the unbounded potential in a Schrödinger equation from the associated Dirichlet-to-Neumann map (abbreviated to DN map in this text). We establish a stability inequality for potentials belonging to $L^n$, where $n\ge 3$ is the dimension of the space. Next, we prove a uniqueness result for potentials in $L^{n/2}$, $n\ge 3$, and apply this uniqueness result to demonstrate a Borg-Levinson type theorem. We use a classical ap…
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.
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。
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 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.
In this article we prove a reducibility result for the linear Schrödinger equation on the sphere $\mathbb{S}^{n}$ with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than 1/2 and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. We notice that our result does not requires the use of the pseudo-differential calculus.
We prove controllability of the Schrödinger equation in $\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schrödinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schrödinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type …
In this article, we construct a global martingale solution to a general nonlinear Schrödinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like $\mathbb{R}^d$, non-compact Riemannian manifolds, and unbounded domains in $\mathbb{R}^d$ with different boundary conditions. The initial value belongs to the energy space $H^1$ and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood-Paley-…
We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.
Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential
本文研究格点上带无界非负势的离散 Schrödinger 算子的半经典波动方程，证明其纯离散谱、Cauchy 问题适定性，并在 ℏ→0 时恢复经典与极弱解。
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.
In this paper, we use the variational method to find the normalized solutions of the quadratic coupled three wave Schrodinger equation with asymmetric coercive potential. We prove the existence of solutions for the system with 2-norm constraints.
A note on the fourth-order Schrodinger equation with spatially growing inhomogeneous source term
本文研究了一个非线性双调和薛定谔方程，考虑了非均匀项的影响，发展了小数据的局部和全局理论，并在更低正则性下发展了局部理论。
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 consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. We prove Hölder stability by choosing the Dirichlet boundary condition suitably.
Error bounds for Physics Informed Neural Networks in Nonlinear Schrödinger equations placed on unbounded domains
针对无界区域上的次临界非线性薛定谔方程，提出新PINNs方法，证明能量与Strichartz范数误差界，并应用于行波、呼吸子与孤子。
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…
The initial value problem for some coupled nonlinear Schrodinger system with unbounded potential is investigated. In the defocusing case, global well-posedness is obtained. For the focusing sign, existence of global and non global solutions is discussed via potential well method. Moreover, existence of ground state and instability of standing waves are proved.
Normalized positive solutions for Schrödinger equations with potentials in unbounded domains
本文研究无界区域中带非负位势的薛定谔方程在给定 $L^2$ 范数约束下正解的存在性，在 $L^q$ 范数小或位势满足衰减条件时证明束缚态解存在，并指出除自治情形外不存在基态解。
Multiple Normalized Solutions to a Class of Modified Quasilinear Schrodinger Equations Schrodinger Equations
本文研究高维拟线性薛定谔方程在质量约束下正解的存在性、非存在性与多重性，通过对偶变换和全局分支方法统一处理次临界、临界和超临界非线性，并刻画了解的渐近行为与无界解连续统。
In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a nonexistence result when the weighthas positive integral. In addition, we show some qualitative properties of the first eigenfunction including a sharp decay estimate. Finally, we extend the decay result to the positive solutions of a Schrödinger type equation.
We consider anti-plane shear deformations of an incompressible elastic solid whose reference configuration is an infinite cylinder with a cross section that is unbounded in one direction. For a class of generalized neo-Hookean strain energy densities and live body forces, we construct unbounded curves of front-type solutions using global bifurcation theory. Some of these curves contain solutions with deformations of arbitrarily large magnitude.
Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential
针对三维及以上有界区域中带 Robin 边界条件的 Schrödinger 算子，本文利用特征值与完整或部分边界 Dirichlet 测量组成的谱数据，研究无界势反演的 Borg-Levinson 唯一性与稳定性。
Two parabolic inverse problems for an equation with unbounded zero-order coefficient
该研究针对带无界零阶系数的抛物型方程，证明了边界测量下系数确定的唯一性（覆盖最大类无界零阶系数），建立了单次内部测量下初值确定的对数稳定性不等式，并借助带无界势的Schrödinger方程的新全局定量唯一延拓性得到对应能观性不等式。
The relativistic Vlasov-Maxwell system describes the evolution of a collisionless plasma. The problem of linear instability of this system is considered in two physical settings: the so-called "one and one-half" dimensional case, and the three dimensional case with cylindrical symmetry. Sufficient conditions for instability are obtained in terms of the spectral properties of certain Schrödinger operators that act on the spatial variable alone (and not in full phase space). An important aspect of these conditions is that they do not require any boundedness assumptions on the domains, nor do the…
We consider the unique recovery of a non compactly supported and non periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different context including recovery of some general class of coefficients from measurements on a bounded sub…
We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast …
In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than $(T-t)^{-1}$, the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.
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…
We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potent…
The aim of this work is to study the controllability of the Schrödinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-Δu(t)~~~~~\text{ on }Ω(t) \tag{$\ast$} \end{equation} with Dirichlet boundary conditions, where $Ω(t)\subset\mathbb{R}^N$ is a time-varying domain. We prove the global approximate controllability of \eqref{eq_abstract} in $L^2(Ω)$, via an adiabatic deformation $Ω(t)\subset\mathbb{R}$ ($t\in[0,T]$) such that $Ω(0)=Ω(T)=Ω$. This control is strongly based on the Hamiltonian structure of \eqref{eq_abstract} provided by [18], which enables the use of adiabatic motio…
We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schrödinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term, in order to weaken the necessary assumptions to obtain a linking structure to the problem, for instance to allow zero being in the spectrum or the nonlinearity being sign-changing. Our main difficulty is to overcome the lack of monotonicity on the nonlinear term, as well, as the lack of compactness since the domain is unbounded. W…
This paper concerns the homogenization of Schrodinger equations for non-crystalline matter, that is to say the coefficients are given by the composition of stationary functions with stochastic deformations. Two rigorous results of so-called effective mass theorems in solid state physics are obtained: a general abstract result (beyond the classical stationary ergodic setting), and one for quasi-perfect materials (i.e. the disorder in the non-crystalline matter is limited). The former relies on the double-scale limits and the wave function is spanned on the Bloch basis. Therefore, we have extend…
Iterative operator-splitting methods for unbounded operators: Error analysis and examples
本文提出并分析了一种针对无界算子的迭代算子分裂方法，推导了误差界，并应用于双曲和抛物型方程以及薛定谔方程。
Standing Waves for Schrödinger Equations with Kato-Rellich potentials
本文利用抛物半流的几何拓扑分析、尾部估计及无界算子谱理论，结合 Conley 指数理论，证明了具有 Kato-Rellich 势的非线性 Schrödinger 方程在共振与非共振情形下的驻波存在性。
We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space, uniformly with respect to the time variable. We show a general exponential control of first order derivatives and momenta, which yields a double exponential bound for higher Sobolev norms and momenta. On the other hand, we show that if the potential is an isotropic harmonic potential with a time dependent frequency which decays sufficiently fast, then Sobolev no…
We are concerned with the optimal constants: in the Korn inequality under tangential boundary conditions on bounded sets $Ω\subset \mathbb{R}^n$, and in the geometric rigidity estimate on the whole $\mathbb{R}^2$. We prove that the latter constant equals $\sqrt{2}$, and we discuss the relation of the former constants with the optimal Korn's constants under Dirichlet boundary conditions, and in the whole $\mathbb{R}^n$, which are well known to equal $\sqrt{2}$. We also discuss the attainability of these constants and the structure of deformations/displacement fields in the optimal sets.
Let (M,g) be a n-dimensional compact Riemannian manifold. We consider the magnetic deformations of semiclassical Schrodinger operators on M for a family of magnetic potentials that depends smoothly on $k$ parameters $u$, for $k \geq n$, and satisfies a generic admissibility condition. Define the deformed Schrodinger eigenfunctions to be the $u$-parametrized semiclassical family of functions on M that is equal to the unitary magnetic Schrodinger propagator applied to the Schrodinger eigenfunctions. The main result of this article states that the $L^2$ norms in $u$ of the deformed Schrodinger ei…
We study the inverse problem of determining a magnetic Schrödinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even p…
We study the reducibility of a Linear Schrödinger equation subject to a small unbounded almost-periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity and on the frequency of the almost-periodic perturbation, we prove that such an equation is reducible to constant coefficients via an analytic almost-periodic change of variables. This implies control of both Sobolev and Analytic norms for the solution of the corresponding Schrödinger equation for all times.
We extend some previous results for the damped wave equation in bounded domains in Euclidean spaces to the unbounded case. In particular, we show that if the damping term is of the form $αa$ with bounded $a$ taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive $α$. In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrodinger operators to the unbounded case, which we believe to be of interest in their own right.
This analysis is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, is corrected with a polarizability term, involving the field induced dipole moment. Sufficient conditions for approximate controllability are given. For transfers between eigenstates of the free Hamiltonian, the control laws are explicitly given. The results apply also for unbounded or non-regular potentials.
We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost estimate which improves upon earlier results. It is particularly interesting when combined with the earlier mentioned spectral inequalities since it yields sharp control cost bounds in several asymptotic regimes. We also show that control problems on unbounded domains can be approximated by corresponding problems on a sequence of bounded domains forming an exhausti…
This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of "admissible" unbounded linear operators (not necessarily closed or closable). This is…
We consider elliptic equations of Schrödinger type with a right-hand side fixed and with the linear part of order zero given by a potential V . The main goal is to study the optimization problem for an integral cost depending on the solution uV , when V varies in a suitable class of admissible potentials. These problems can be seen as the natural extension of shape optimization problems to the framework of potentials. The main result is an existence theorem for optimal potentials, and the main difficulty is to work in the whole Euclidean space Rd, which implies a lack of compactness in several…
Optimal backward uniqueness and polynomial stability of second order equations with unbounded damping
针对无界阻尼二阶方程，给出排除有限时间熄灭的最优条件，通过控制估计与伸缩方法导出显式衰减率，并应用于奇异阻尼、重力水波和欧拉-伯努利板等系统。
Under natural energy and decay assumptions, we derive a priori estimates for solutions of a Schrodinger-Newton type of equation with critical exponent. On one hand, such an equation generalizes the traditional Schrodinger-Newton and Choquard equations, while, on the other hand, it is naturally related to problems involving scalar curvature and conformal deformation of metrics.
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