A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

A family of predictor-corrector exponential Numerov-type methods is developed for the numerical integration of the one-dimensional Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. The new methods are very simple and integrate more exponential functions than both the well-known fourth-order Numerov-type exponentially fitted methods and the sixth algebraic order Runge-Kutta-type methods. Numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above.     

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.     

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

     

On the optimal control problem for Schrödinger equation with complex potential

The existence and uniqueness for the solution of the problem of determining the v(x,t) potential in the Schrödinger equation from the measured final data ψ(x,T)=y(x) is investigated. For the objective functional , it is proven that the problem has at least one solution for α?0, and has a unique solution for α>0. The necessary condition for solvability the problem is stated as the variational principle.     

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.     

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.     

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.     

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.     

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if

     

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.     

Dimension of divergence sets for the pointwise convergence of the Schrödinger equation

In this note, we consider the pointwise convergence along curves for the Schrödinger equation and obtain estimates for the capacitary dimension of divergence sets which extend our previous result in [6].     

Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation

We study existence results for a nonlinear Schrödinger equation at resonance. The nonlinearity is assumed to change sign, be unbounded but sublinear with a power like growth at infinity. Under a suitable coercivity assumption on the primitive of the nonlinear term on the kernel of the Schrödinger operator, we prove the existence of at least one solution.     

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.     

Some new Strichartz estimates for the Schrödinger equation

We deal with fixed-time and Strichartz estimates for the Schrödinger propagator as an operator on Wiener amalgam spaces. We discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones. As an application, we present a result on the wellposedness of the linear Schrödinger equation with a rough time-dependent potential.     

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.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.