Marius Beceanu
This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem.
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In this paper we develop the classical multiplier technique to prove a virial identity and smoothing estimates (in a perturbative setting) for the electromagnetic variable coefficients Schroedinger equation.
We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.
This paper has been withdrawn by the author due to a crucial error in the Proof of Theorem 0.3
We consider the Schroedinger operator in R^3 with N point interactions placed at Y=(y_1, ... ,y_N), y_j in R^3, of strength a=(a_1, ... ,a_N). Exploiting the spectral theorem and the rather explicit expression for the resolvent we prove a (weighted) dispersive estimate for the corresponding Schroedinger flow. In the special case N=1 the proof is directly obtained from the unitary group which is known in closed form.
In this paper, we study some modified linear restriction estimates of the dynamics generated by Schroedinger operator on metric cone $M$, where the metric cone $M$ is of the form $M=(0,\infty)_r\timesΣ$ with the cross section $Σ$ being a compact $(n-1)$-dimensional Riemannian manifold $(Σ,h)$ and the equipped metric is $g=\mathrm{d}r^2+r^2h$. Assuming the initial data possesses additional regularity in angular variable $θ\inΣ$, we show some linear restriction estimates for the solutions. As applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initi…
In this paper, we consider the maximal estimates for the solution to an initial value problem of the linear Schroedinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special variable coefficient Schroedinger equation with initial data $u_0\in H^{s}(\R^n)$ for $s>1/2$ or radial initial data $u_0\in H^{s}(\R^n)$ for $s\geq1/4$ and the solution does not converge when $s<1/4$.
In space dimension $n\geq3$, we consider the magnetic Schrödinger Hamiltonian $H=-(\nabla-iA(x))^2$ and the corresponding Schrödinger equation i\partial_tu+Hu=0. We show some explicit examples of potentials $A$, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrödinger admissibility.
We show that the time evolution of the operator $H = -Δ+ i(A \cdot \nabla + \nabla \cdot A) + V$ in R^3 satisfies Strichartz and smoothing estimates under suitable smoothness and decay assumptions on A and V but without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.
We prove a general dispersive estimate for a Schroedinger type equation on a product manifold, under the assumption that the equation restricted to each factor satisfies suitable dispersive estimates. Among the applications are the two-particle Schroedinger equations on Euclidean spaces, and the nonlinear Schroedinger equation on the product of two real hyperbolic spaces.
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequali…
We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.
Here, the Morgan type uncertainty principle and unique continuation properties of abstract Schredinger equations with time dependent potentials are obtained in Hilbert space valued function classes. The equations include linear operator in abstract Hilbert spaces H dependent on space variables. So, by selecting appropriate spaces H and operators, we derive unique continuation properties for numerous classes of Schrödinger type equations and its systems, which occur in a wide variety of physical systems.
We consider the Schrödinger type operator ${\mathcal A}=(1+|x|^α)Δ-|x|^β$, for $α\in [0,2]$ and $β\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in $L^p(\R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t\ge 0}$. For $α\in [0,2)$ and $β\ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${\mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operat…
The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schroedinger (NLS) equation with data u_0 in the function space ^L^r:=^H^r_0, where for s \in R the spaces ^H^r_s are defined by the norms ||u_0||_{^H^r_s}:=||^u_0||_{L^r'_ξ}, 1/r + 1/r'=1. Similar arguments, combined with a…
We consider in the whole plane the Hamiltonian coupling of semilinear Schroedinger equations which have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to the singularly perturbed system when the external potential stays bounded away from zero. Namely, as the adimens…
L^p Boundedness of the Scattering Wave Operators of Schroedinger Dynamics with Time-dependent Potentials and applications
本文通过新的消去引理证明了含时位势线性薛定谔方程波算子的L^p有界性，并应用于非线性色散方程，得到Hartree方程全局存在性及自由通道波算子的存在性。
We consider time-dependent Schroedinger equations in one dimension with double well potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest eigenvalues then, in the semiclassical limit, we show that the reduction of the time-dependent equation to a 2-mode equation gives the dominant term of the solution with a precise estimate of the error. By means of this stability result we are able to prove the destruction of the beating motion for large enough nonlinearity.
We prove mixed norm space-time estimates for solutions of the Schroedinger equation, with initial data in $L^p$ Sobolev or Besov spaces, and clarify the relation with adjoint restriction.
We prove an optimal 4G Theorem for the Gaussian kernel. We also propose a new general method of estimating Schroedinger perturbations of transition densities, and give applications to the Gaussian kernel.
We present some old and new results on dispersive estimates for Schroedinger equations.
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.
We prove a dispersive estimate for the one-dimensional Schroedinger equation, mapping between weighted $L^p$ spaces with stronger time-decay ($t^{-3/2}$ versus $t^{-1/2}$) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that $(1+|x|)^3 V$ be integrable and $-Δ+ V$ not have a resonance at zero energy. If a resonance is present (for example in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.
We establish a dispersive estimate (with a decay of 1/t), valid for all times, for the Schroedinger evolution on a non-compact 2-dimensional manifold with a trapped geodesic.
We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential enable us to also treat the case of small semi-linear perturbations. In both the linear and the nonlinear instances, we prove that, on average, energy remains bounded and solutions scatter.
In this paper I prove a L^p-L^p' estimate for the solutions of the one-dimensional Schroedinger equation with a potential in L^1_gamma where in the generic case gamma > 3/2 and in the exceptional case (i.e. when there is a half-bound state of zero energy) gamma > 5/2. I use this estimate to construct the scattering operator for the nonlinear Schroedinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonl…
We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate in the argument is a Morawetz-type inequality for the solution $ϕ$. This estimate bounds $\|ϕ(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(ϕ(x,t))^4}{|x|} dx dt
We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular point for the perturbed resolvent.
A version of the binary Darboux transformation is constructed for non-stationary Schroedinger equation in dimension $k+1$, where $k$ is the number of space variables, $k \geq 1$. This is an iterated GBDT version. New families of non-singular and rational potentials and solutions are obtained. Some results are new for the case that $k=1$ too. A certain generalization of a colligation introduced by M.S. Livsic and of the $S$-node introduced by L.A. Sakhnovich is successfully used in our construction.
Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi Golden rule. The exponential decay rate of resonances is addressed; its uniqueness in the time dependent picture is shown is certain cases. The relation to the existence of meromorphic continuation of the properly weighted Green's function to time dependent resonance is further elucidated, by giving an equivalent time dependent asymptotic expansion of the solu…
We study Schroedinger operators with Robin boundary conditions on exterior domains in $\R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.
We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schroedinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm.
We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the …
Propagation of Regularity for Schroedinger Equations with Time Dependent Potentials
研究含一般时间依赖势的薛定谔方程，证明对于空间局部化势，存在正则性传播且高阶Sobolev范数一致有界，通过直接传播估计处理非散射部分。
We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems.
We prove a dispersive estimate for the evolution of Schroedinger operators $H = -Δ+ V(x)$ in ${\mathbb R}^3$. The potential is allowed to be a complex-valued function belonging to $L^p(\R^3)\cap L^q(\R^3)$, $p < \frac32 < q$, so that $H$ need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of $H$ exist anywhere within the interval $[0,\infty)$ and that eigenfunctions at zero (including generalized eigenfunctions) decay rapidly enough to be integrable.
In this paper we study the maximal regularity property for non-autonomous evolution equations $\partial_t u(t)+A(t)u(t)=f(t), u(0)=0.$ If the equation is considered on a Hilbert space $H$ and the operators $A(t)$ are defined by sesquilinear forms $ a(t,.,.)$ we prove the maximal regularity under a Holder continuity assumption of $t \to a(t,.,.)$. In the non-Hilbert space situation we focus on Schrodinger type operators $A(t):= -Δ+ m(t, .)$ and prove $L^p-L^q$ estimates for a wide class of time and space dependent potentials $m$.
For $α>1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-Δ)^{α/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.
The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schroedinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey-Stewartson and Gross-Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schroedinger equations in spat…
In this paper, we study the decoherence of a wave described by the solution to a Schroedinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence behaviors is a properly rescaled Wigner transform of the solution of the random Schroedinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.
This paper proves endpoint Strichartz estimates for the linear Schroedinger equation in $R^3$, with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large in norm. The potential is also subjected to a time-dependent rescaling, with a non-differentiable dilation parameter. We use the Strichartz estimates to prove the non-dispersion of bound states, when the path is small in norm, as well as boundedness of energy. We also include a sample nonlinear application of the linear results.
This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where the power of nonlinearity is somewhat smaller than the pseudoconformal power and the initial data belong to the scale-invariant homogeneous Sobolev space. In spite of the negative-order differentiability of initial data the nonlinear Schroedinger equation has global in time solutions provided that the initial data have the small norm. The key ingredient in t…
We study local in time Strichartz estimates for the Schroedinger equation associated to long range perturbations of the flat Laplacian on the euclidean space. We prove that in such a geometric situation, outside of a large ball centered at the origin, the solutions of the Schroedinger equation enjoy the same Strichartz estimates as in the non perturbed situation. The proof is based on the Isozaki-Kitada parametrix construction. If in addition the metric is non trapping, we prove that the Strichartz estimates hold in the whole space.
We prove some uniform in $ε$ a priori estimates for solutions of the equation $$(\nabla-iA)^2u-V(x)u+(λ\pm iε)u=f, λ\geq0, ε\neq0.$$ The estimates are obtained in terms of Morrey-Campanato norms, and can be used to prove absence of zero-resonances, in a suitable sense, for electromagnetic Hamiltonians. Precise conditions on the size of the \textit{trapping component} of the magnetic field and the non repulsive component of the electric field are given.
We prove Strichartz estimates for the Schroedinger operator $H = -Δ+ V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies,…
We study Schroedinger's equation with a potential moving along a Brownian motion path. We prove a RAGE-type theorem and Strichartz estimates for the solution on average.
I consider 4-dimensional Schrödinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(μ^{-1}h^{-3})$ and the principal part $\asymp h^{-4}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field. For general potentials remainder estimate $O(μ^{-1}h^{-3}+μ^2h^{-2})$ is achieved.
We prove the sharp L^1-L^{\infty} time-decay estimate for the 2D-Schroedinger equation with a general family of scaling critical electromagnetic potentials.
We give a short description of the proof of asymptotic-completeness for NLS-type equations, including time dependent potential terms, with radial data in three dimensions. We also show how the method applies for the two-body Quantum Scattering case.
We consider 2-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider 3-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to $O(h^{-1}|\log h|)$ and with the principal part $\asymp μh^{-2}$.
We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere $\mathbb{S}^2$ and more generally on Zoll surfaces, we prove that, for $s>1/4$, NLS is uniformly well-posed in $H^s$, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. On étudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte.Sur la sphère $\mathbb{S}^2$ et plus généralement sur toute surface de Zoll, on démontre que pour $s>1/4$, NLS est uniformément bien posée dans $H^s$, ce qui est opt…
We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any specific properties of NLS. Rather, it is axiomatic on the linear level, and our results are obtained from four assumptions (which are of course motivated by NLS). This work is in three dimensions.
We prove a sharp Mihlin-Hormander multiplier theorem for Schroedinger operators $H$ on $\R^n$. The method, which allows us to deal with general potentials, improves Hebisch's method relying on heat kernel estimates for positive potentials. Our result applies to, in particular, the negative Poeschl-Teller potential $V(x)= -ν(ν+1) \sech^2 x $, $ν\in \N$, for which $H$ has a resonance at zero.
Global in time dispersive estimates for the Schroedinger and wave evolutions are obtained on manifolds with conical ends whose Hamiltonian flow exhibits trapping. This paper deals with the case of initial data with fixed "nonzero angular momentum".
We consider time-dependent nonlinear Schroedinger equations subject to smooth, lattice-periodic potentials plus additional confining potentials, slowly varying on the lattice scale. After an appropriate scaling we study the homogenization limit for vanishing lattice spacing. Assuming well prepared initial data, the resulting effective dynamics is governed by a homogenized nonlinear Schroedinger equation with an effective mass tensor depending on the initially chosen Bloch eigenvalue. The given results rigorously justify the use of the effective mass formalism for the description of Bose-Einste…
Consider the focusing cubic semilinear Schroedinger equation in R^3 i \partial_t ψ+ Δψ+ | ψ|^2 ψ= 0. It admits an eight-dimensional manifold of special solutions called ground state solitons. We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant. Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a soliton and radiation. Conve…
Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we consider the associated wave operators W_+, W_- defined as the strong L^2 limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the wave operators are bounded operators on L^p for all 1<p<\infty, provided (1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a resonance. For p=\infty we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.
The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Schroedinger operators defined on the two-dimensional disk with a radially symmetric magnetic field and radially symmetric electric potential.
We obtain explicit solutions for the density $φ_T$ of the first-time $T$ that a one-dimensional Brownian process $B$ reaches the twice, continuously differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for all $t\in \mathbb{R}^+$. We do so by finding the expected value of some functionals of a 3-dimensional Bessel bridge $\tilde{X}$ and exploiting its relationship with first-passage time problems as pointed out by Kardaras (2007). It turns out that this problem is related to Schrödinger's equation with time-dependent linear potential, see Feng (2001).
The large time asymptotics of nonlinear multichannel Schroedinger equations
该研究针对带一般空间局域相互作用的非线性多通道薛定谔方程，通过相空间分析与新的外区域先验传播估计，证明径向对称且H¹一致有界的解强渐近于自由波加弱局域化解，可统一分析线性含时势与非线性相互作用。
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