Some lower bounds for solutions of Schrodinger evolutions

Concentration and non-concentration for the Schrödinger evolution on Zoll manifolds

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential to the Laplacian on the sphere results on the existence of geodesics $γ$ such that $δ_γ$ cannot be obtained as a semiclassical measure for some sequence of eigenfunctions. We also show that the same phenomenon occurs for the free Lapl…

On construction of solutions of evolutionary Non Linear Schrodinger equation

In this work we present an application of a theory of vessels to solution of the evolutionary Non Liner Schrodinger (NLS) equation. The classes of functions for which the initial value problem is solvable relies on the existence of an analogue of the inverse scattering theory for the usual NLS equation. This approach is similar to the classical approach of Zackarov-Shabbath for solving of evolutionary NLS equation, but has an advantage of simpler formulas and new techniques and notions to construct solutions of the evolutionary NLS equation.

On the time evolution of Wigner measures for Schrodinger equations

In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the …

Lower bounds for estimates of the Schrödinger maximal function

We give new lower bounds for $L^p$ estimates of the Schrödinger maximal function by generalizing an example of Bourgain.

Pointwise Lower Bounds on the Heat Kernels of Higher Order Schrödinger Operators

We apply Davies' method for obtaining pointwise lower bounds on the heat kernels of higher-order differential operators to obtain pointwise lower bounds in the presence of a polynomialy bounded potential.

Exponential Lower Bounds for Quasimodes of Semiclassical Schrödinger Operators

We prove quantitative unique continuation results for the semiclassical Schrodinger operator on smooth, compact domains. These take the form of exponentially decreasing (in h) local L^{2} lower bounds for exponentially precise quasimodes. We also show that these lower bounds are sharp in h, and that, moreover, the hypothesized quasimode accuracy is also sharp.

Schrödinger evolution of superoscillations with $δ$- and $δ'$-potentials

In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schrödinger equation with $δ$- and $δ'$-potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converges in the topology of the entire function space $A_1(\mathbb{C})$. Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave …

Nonlinear Schrödinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres

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.

Upper and lower $L^2$-decay bounds for a class of derivative nonlinear Schrödinger equations

We consider the initial value problem for cubic derivative nonlinear Schrödinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as $t\to +\infty$. Furthermore, we find that this $L^2$-decay rate is optimal by giving a lower estimate of the same order.

Improved Lower Bound for Analytic Schrödinger Eigenfunctions in Forbidden Regions

The point of this paper is to improve the reverse Agmon estimate discussed in \cite{TW} with assuming that the Schrodinger operator $P(h) = - h^2 Δ_g + V - E(h)$, $E(h)\to E$ as $h\to 0^+$, is analytic on a compact, real-analytic Riemannian manifold $(M,g)$. In this paper, by considering a Neumann problem with applying Poisson representation and exterior mass estimates on hypersurfaces, we can prove an improved reverse Agmon estimate on a hypersurface.

A sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearity

Let $T_ε$ be the lifespan for the solution to the Schrödinger equation on $\mathbb{R}^d$ with a power nonlinearity $λ|u|^{2θ/d}u$ ($λ\in \mathbb{C}$, $0<θ<1$) and the initial data in the form $εφ(x)$. We provide a sharp lower bound estimate for $T_ε$ as $ε\to +0$ which can be written explicitly by $λ$, $d$, $θ$, $φ$ and $ε$. This is an improvement of the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].

Mass Propagation for Electromagnetic Schrödinger Evolutions

We investigate the validity of gaussian lower bounds for solutions to an electromagnetic Schrödinger equation with a bounded time-dependent complex electric potential and a time-independent vector magnetic potential. We prove that, if a suitable geometric condition is satisfied by the vector potential, then positive masses inside of a bounded region at a single time propagate outside the region, provided a suitable average in space-time cylinders is taken.

Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball

We establish new results for the radial nonlinear wave and Schrödinger equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from \cite{B-T1,B-T2} and \cite {T1,T2}.

Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations

In this paper we consider linear, time dependent Schrödinger equations of the form $i \partial_t ψ= K_0 ψ+ V(t) ψ$, where $K_0$ is a positive self-adjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant. We give a strategy to construct bounded perturbations $V(t)$ such that the Hamiltonian $K_0 + V(t)$ generates unbounded orbits. We apply our abstract construction to three cases: (i) the Harmonic oscillator on $\mathbb R$, (ii) the half-wave equation on $\mathbb T$ and (iii) the Dirac-Schrödinger equation on the sphere. In each case, $V(t)$ is a smooth and p…

Notes on Hardy's Uncertainty Principle for the Wigner distribution and Schrödinger evolutions

We consider Schrödinger equations with real quadratic Hamiltonians, for which the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we prove a uniqueness result for such Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach reproduces known, sharp results for the free Schrödinger equation and the harmonic oscillator, and we also present an explicit scheme for qu…

高维周期薛定谔极大算子的尖锐下界

Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions

本文证明了高维环面上薛定谔极大函数的尖锐下界，结合已有充分性结果，完全解决了周期薛定谔方程Carleson问题，并反驳了Miao-Yuan-Zhao猜想，方法基于阿贝尔簇上的复乘法。

Uniqueness for discrete Schrodinger evolutions

We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general real-valued time-dependent bounded potentials. In the latter case the result is not optimal.

$N$体Schrödinger演化方程的混合正则性与稀疏网格逼近

Mixed regularity and sparse grid approximations of $N$-body Schrödinger evolution equation

本文分析了含时$N$体电子系统的波函数混合正则性，据此开发了稀疏网格逼近（含GTO格式）以降低计算复杂度，并在氦原子和氢分子上验证了其相比全网格离散化的高效性。

On sharp lower bound of the spectral gap for a Schrödinger operator and some related results

In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at same estimates by the `double coordinate' approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral…

关于 Hilbert 空间上的分数阶抽象 Schrödinger 型发展方程及其在分数阶色散方程中的应用

On the fractional abstract Schrodinger-type evolution equations on the Hilbert space and its applications to the fractional dispersive equations

本文基于自伴算子谱定理和 Mittag-Leffler 函数渐近行为，证明了 Hilbert 空间上时间分数阶抽象 Schrödinger 型发展方程及分数阶色散方程的局部和全局适定性，并给出了线性方程温和解的 Hölder 正则性与渐近行为。

Talbot effect for the cubic nonlinear Schrödinger equation on the torus

We study the evolution of the one dimensional periodic cubic Schrödinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. For rational times the solution is a linear combination of finitely many translates of the initial data. Such a dichotomy was first observed by Talbot in an optical experiment performed in 1836. In this paper we prove that a similar phenomenon occurs in the case of the NLS equation.

Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions

Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian systems of PDEs, as well as certain nonlinear Schrodinger evolutions. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as "periodic orbits up to an isometry" for evolution equations that may not have periodic solutions. In the process, we introdu…

On the logarithmic Schrodinger equation

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

有界 Lipschitz 函数上方区域中 Schrödinger 算子的热核估计

Heat Kernel Estimates for Schrödinger Operators in the Domain Above a Bounded Lipschitz Function

本文研究有界 Lipschitz 图像上方区域内 Schrödinger 算子的 Dirichlet 热核，并在势函数远离边界呈超二次衰减时建立匹配的上下界。

散焦参数非线性 Schrödinger 系统中的曲率与混沌

Curvature and Chaos in the Defocusing Parameteric Nonlinear Schrodinger System

本文研究散焦参数非线性 Schrödinger 系统中弯曲暗孤子前沿的法向速度，揭示参数强度可使其由曲率驱动流转变为表面扩散正则化的逆曲率运动，并导致界面增长、自交及混沌。