Optimal regularity for Schrödinger equations

In this paper we study the regularity theory for the Schrödinger equations under proper conditions. Furthermore, it will be verified that these conditions are optimal.     

Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential

This paper is concerned with the nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose–Einstein condensate under the magnetic trap. By combining the best constant of Gagliardo–Nirenberg's inequality with the characteristic of this equation, we derive out a global existence condition for the supercritical equation which coincides with the critical case.     

Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank–Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L norm

We prove global wellposedness for the one-dimensional cubic non-linear Schrödinger equation in a space of distributions which is invariant under Galilean transformations and includes L². This space arises naturally in the study of the restriction properties of the Fourier transform to curved surfaces. The L^p bounds, p≠2, for the extension operator, dual to the restricition one, plays a fundamental role in our approach.     

On the Dirac delta as initial condition for nonlinear Schrödinger equations

In this article we will study the initial value problem for some Schrödinger equations with Dirac-like initial data and therefore with infinite L² mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so-called Hasimoto transformation.     

Multi solitary waves for nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation in for any d1, with a nonlinearity such that solitary waves exist and are stable. Let R_k(t,x) be K arbitrarily given solitary waves of the equation with different speeds v₁,v₂,…,v_K. In this paper, we prove that there exists a solution u(t) of the equation such that

     

On Schrödinger operators with multipolar inverse-square potentials

Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.     

A multisymplectic integrator for the periodic nonlinear Schrödinger equation

A multisymplectic integrator for the periodic nonlinear Schrödinger equation is presented in this paper. Its accuracy is proved. By introducing a norm, we investigate its nonlinear stability. We also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method.     

Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices

We consider C=A+B where A is selfadjoint with a gap (a,b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV)^d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.     

Discrete artificial boundary conditions for nonlinear Schrödinger equations

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.     

A four-step method for the numerical solution of the Schrödinger equation

A new four-step exponentially-fitted method is developed in this paper. The expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue Schrödinger equation (i.e., equivalent to positive energy).     

Computing Schrödinger propagators on Type-2 Turing machines

We study Turing computability of the solution operators of the initial-value problems for the linear Schrödinger equation u_t=iΔu+φ and the nonlinear Schrödinger equation of the form iu_t=-Δu+mu+|u|²u. We prove that the solution operators are computable if the initial data are Sobolev functions but noncomputable in the linear case if the initial data are L^p-functions and p≠2. The computations are performed on Type-2 Turing machines.