Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C^∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?     

Unique continuation property for a higher order nonlinear Schrödinger equation

We prove that, if a sufficiently smooth solution u to the initial value problem associated with the equation

     

On minimal support properties of solutions of Schrödinger equations

In this paper we obtain minimal support properties of solutions of Schrödinger equations. We improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small. We also use these properties to obtain some new results on unique continuation for the Schrödinger operator.     

Unique continuation for nonnegative solutions of Schrödinger type inequalities

We prove a sharp unique continuation theorem for nonnegative H2,1 solutions of the differential inequality |Δu(x)|?C⁻²|x−x₀||u(x)| which vanish of finite order at x₀.     

A counter example to Strichartz estimates for the inhomogeneous Schrödinger equation

We disprove Strichartz estimates for the solution of the inhomogeneous Schrödinger equation in a certain range of the Lebesgue exponents values.     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials

Let u be the weak solution to the degenerate Schrödinger equation with singular coefficients in Lipschitz domain as following

−div(w(x)A(x)∇u(x))+V(x)u(x)w(x)=0,     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).     

Improved inhomogeneous Strichartz estimates for the Schrödinger equation

We study inhomogeneous Strichartz estimates for the Schrödinger equation for dimension n?3. Using a frequency localization, we obtain some improved range of Strichartz estimates for the solution of inhomogeneous Schrödinger equation except dimension n=3.     

Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation

     

A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q?2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0<t<1|eitΔf| is bounded from Hs(Rn) to when s>s₀ if and only if it is bounded from Hs(Rn) to L²(Rn) when s>2s₀. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R²) to L²(R²) when s>3/4.     

Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation

In this article we establish the bilinear estimates corresponding to the 1D and 2D NLS with a quadratic nonlinearity , which imply the local well-posedness of the Cauchy problem in Hs for s?−1 in the 1D case and for s>−1 in the 2D case. This is a continuation of our study [N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity , Commun. Pure Appl. Anal. 7 (2008) 1123-1143] on the 1D NLS with nonlinearity . Previous papers by Kenig, Ponce and Vega, and Colliander, Delort, Kenig and Staffilani established local well-posedness for s>−3/4 in 1D and in 2D, respectively, and when the nonlinearity is restricted to cu², papers by Bejenaru and Tao, and Bejenaru and De Silva improved these results to s?−1 in 1D and s>−1 in 2D. The bilinear estimate for 2D also yields an improvement on the growth rate of Sobolev norms of finite energy global-in-time solutions to the 2D cubic NLS.     

Locating peaks of a Schrödinger equation with sign-changing nonlinearity

In this paper, we study the concentration phenomenon of a positive ground state solution of a nonlinear Schrödinger equation on R^N. The coefficient of the nonlinearity of the equation changes sign. We prove that the solution has a maximum point at x₀∈Ω⁺={x∈R^N:Q(x)>0} where the energy attains its minimum.     

An optimal control problem for two-dimensional Schrödinger equation

In this paper we consider an optimal control problem controlled by three functions which are in the coefficients of a two-dimensional Schrödinger equation. After proving the existence and uniqueness of the optimal solution, we get the Frechet differentiability of the cost functional using Hamilton-Pontryagin function. Then we state a necessary condition to an optimal solution in the variational inequality form using the gradient.     

Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials

In this paper we investigate the existence of positive solutions for the quasilinear Schrödinger equation:

−Δu+V(x)u−Δ(u²)u=g(u),$-\mathrm{\Delta }u+V(x)u-\mathrm{\Delta }\left(u^2\right)u=g(u),$

in R^N${\mathbb{R}}^N$, where N?3$N?3$, g has a quasicritical growth and V is a nonnegative potential, which can vanish at infinity.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Endpoint Strichartz estimates for the magnetic Schrödinger equation

We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n?3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.     

p-Laplace Schrdinger热方程的椭圆型梯度估计

本文对p-Laplace Schr(o|¨)dinger热方程正连续弱解做了椭圆型梯度估计;作为应用,本文得到了一个关于p-Laplace算子的Liouville型结果;并证明了关于p-LaplaceSchr(o|¨)dinger热方程正连续弱解的Harnack不等式.