Magnetic virial identities,weak dispersion and Strichartz inequalities

We show a family of virial-type identities for the Schrödinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension n ≥ 3, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trapping electromagnetic potential with almost Coulomb decay.     

Strichartz estimates for the Schrödinger propagator for the Laguerre operator

We obtain homogeneous Strichartz estimate for the Schrödinger propagator e $^{-itL_{\alpha}}$ for the Laguerre operator L _α on ${\mathbb R}_+^n$ . We follow regularization technique as introduced in J. Funct. Anal. 224(2) (2005) 371–385. We also establish inhomogeneous Strichartz estimates for different admissible pairs.     

Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

We prove well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range \([1, 1 + 4/d)\) of admissible exponents in the nonlinear part (where \(d\) is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

In this paper we consider the Schrödinger type operators \(H_2=(-\Delta)^2 +V^2\), where the nonnegative potential V belongs to the reverse Hölder class \(B_{q_{_1}}\) for \(q_{_1}\geq \frac{n}{2}, n\geq 5\). The L ^p and weak type (1, 1) estimates of higher order Riesz transform \(\nabla^2H^{-\frac{1}{2}}_2 \) related to Schrödinger type operators H ₂ are obtained. In particular, \(\nabla^2H^{-\frac{1}{2}}_2 \) is a Calderón-Zygmund operator if V?∈?B _2n or \(V\in B_\frac{n}{2}\) and there exists a constant C such that V(x)?≤?Cm(x,V)².     

On Schrödinger Propagator for the Special Hermite Operator

We establish a Strichartz type estimate for the Schrödinger propagator e ^it? for the special Hermite operator ? on ? ⁿ . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2.     

Carleman Estimates for the Schrdinger Equation and Applications to an Inverse Problem and an Observability Inequality

The authors prove Carleman estimates for the Schrdinger equation in Sobolev spaces of negative orders, and use these estimates to prove the uniqueness in the inverse problem of determining Lp-potentials. An L2-level observability inequality and unique continuation results for the Schrdinger equation are also obtained.     

Semilinear Schrödinger flows on hyperbolic spaces: scattering in <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

We prove global well-posedness and scattering in H ¹ for the defocusing nonlinear Schrödinger equations

$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H ¹ is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

     

Schrödinger soliton from Lorentzian manifolds

In this paper, we introduce a new notion named as Schrödinger soliton. The so-called Schrödinger solitons are a class of solitary wave solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, then the Schrödinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold M is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton solution to the hyperbolic Ishimori system.     

Spectral properties of the Schrödinger operator with <Emphasis Type="Italic">δ</Emphasis>-distribution

For the one-dimensional Schrödinger operator with δ-interactions, two-sided estimates of the distribution function of the eigenvalues and a criterion for the discreteness of the spectrum in terms of the Otelbaev function are obtained. A criterion for the resolvent of the Schrödinger operator to belong to the class S_p is established.     

Norm convergence of multiple ergodic averages on amenable groups

We consider abstract non-negative self-adjoint operators on L²(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on Rⁿ, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

<Emphasis Type="Italic">q</Emphasis>-Deformed Barut–Girardello <Emphasis Type="Italic">su</Emphasis>(1, 1) coherent states and Schrödinger cat states

We define Schrödinger cat states as superpositions of q-deformed Barut–Girardello su(1, 1) coherent states with an adjustable angle φ in a q-deformed Fock space. We study the statistical properties of the q-deformed Barut–Girardello su(1, 1) coherent states and Schrödinger cat states. The statistical properties of photons are always sub-Poissonian for q-deformed Barut–Girardello su(1, 1) coherent states. For Schrödinger cat states in the cases φ = 0, π/2, π, the statistical properties of photons are always sub-Poissonian if φ = π/2, and the other cases are hard to determine because they depend on the parameters q and k. Moreover, we find some interesting properties of Schrödinger cat states in the limit |z| → 0, where z is the parameter of those states. We also derive that the statistical properties of photons are sub-Poissonian in the undeformed case where π/2 ≤ φ ≤ 3π/2.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from [4]. As a consequence of this, we obtain sharp (up to ε losses) Strichartz estimates for the hyperbolic Schrödinger equation on the torus. Our second main result is an l ² decoupling for nondegenerate curves, which has implications for Vinogradov’s mean value theorem.     

Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x ²,…,x ^p ,exp?(±wx),xexp?(±wx),…,x ^m exp?(±w x)} and the second is based on the exact integration of the functions of the form: {1,x,x ²,…,x ^p ,exp?(±wx),exp?(±2wx),…,exp?(±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrödinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrödinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrödinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrödinger equation. Finally, numerical results for the resonance problem of the radial Schrödinger equation are presented.     

Besov Spaces for the Schrödinger Operator with Barrier Potential

Let H = ?d ²/dx ² + V be a Schrödinger operator on the real line, where \({V=c\chi_{[a,b]}}\) , c > 0. We define the Besov spaces for H by developing the associated Littlewood–Paley theory. This theory depends on the decay estimates of the spectral operator \({{\varphi}_j(H)}\) for the high and low energies. We also prove a Mihlin multiplier theorem on these spaces, including the L ^p boundedness result. Our approach has potential applications to other Schrödinger operators with short-range potentials.     

Decay estimates of discretized Green's functions for Schrdinger type operators

For a sparse non-singular matrix A, generally A~(-1)is a dense matrix. However, for a class of matrices,A~(-1)can be a matrix with off-diagonal decay properties, i.e., |A_(ij)~(-1)| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green's functions for Schr¨odinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter.We verify the decay estimate with numerical results for one-dimensional Schr¨odinger type operators.     

Divergent solutions to the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-supercritical NLS equations

We investigate the nonlinear Schrödinger equation iu _t +Δu+|u| ^p?1 u = 0with 1+ 4/N < p < 1+ 4/N?2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H ¹ and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence t _n → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of ?(1?s _c )Q+ΔQ+Q ^p?1 Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.     

Picard dimension of signed radial Kato measures

The Picard dimension dimμ of a signed local Kato measure μ on the punctured unit ball in R^d, d ≥ 2, is the cardinal number of the set of extremal rays of the convex cone of all continuous solutions u ≥ 0 of the time-independent SchrSdinger equation Δu -- uμ = 0 on the punctured ball 0 〈 ｜｜x｜｜ 〈 1, with vanishing boundary values on the sphere ｜｜x｜｜ = 1. Using potential theory associated with the Schrodinger operator we prove, in this paper, that the dimμ for a signed radial Kato measure is 0, 1 or ＋∞. In particular, we obtain the Picard dimension of locally Holder continuous functions P proved by Nakai and Tada by other methods.