A priori estimates and existence of real-valued periodic solutions to the modified Benjamin-Ono equation with initial data in $H^s$ for $s>1/4$ are proved locally time. The approach relies on frequency dependent time localization, after which dispersive properties from Euclidean space recovered. same regularity results cubic derivative nonlinear Schr\"odinger equation.

A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following spirit of methods, trial test spaces are spanned by non-polynomial complex wave functions that satisfy Schro\"odinger locally on each element mesh. This allows a significant reduction in number degrees freedom comparison full polynomial spaces. ...

The Schr\"odinger equation with a harmonic potential coupled to the Poisson equation, called Schr\"odinger-Newton-Hooke (SNH) system, has been considered in variety of physical contexts, ranging from quantum mechanics general relativity. Our work is directly motivated by fact that SNH system describes nonrelativistic limit Einstein-massive-scalar negative cosmological constant. With this paper ...

Employing a higher order symplectic integration algorithm, we integrate the 1-d nonlinear Schrr odinger equation numerically for solitons moving in an external potential. In particular, we study the scattering oo an interface separating two regions of constant potential, with a linearly rising ramp in the interface region. Transmission coeecients are computed as functions of the height of this ...

In this paper we consider smooth analogues of operators studied in connection with pointwise convergence of the solution of the free Schrr odinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we obtain sharp L 2 ! L 2 bounds. 0. Introduction. We begin this paper by giving motivation for the obj...

The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an in nitedimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. How...

The recovery of the coe cient H(x) in the one-dimensional generalized Schrodinger equation d =dx + kH(x) = Q(x) ; where H(x) is a positive, piecewise continuous function with positive limits H as x ! 1; is studied. The large-k asymptotics of the wavefunctions and the scattering coe cients are analyzed. A factorization formula is given expressing the total scattering matrix as a product of simp...

We study a new type of symmetry for the hydrogen atom involving algebraic families groups parametrized by energy value in time-independent Schr\"odinger equation. construct an family Harish-Chandra modules from solutions equation, and we characterize this family. show that subspaces physical states may be obtained our using Jantzen filtration, relate methods with spectral theory scattering limi...

In this paper we continue our study [DSS20] of the nonlinear Schr\"odinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here prove global 1D NLS lying in $L^{p}$ any $2 < p \infty$, provided is sufficiently smooth. We use complete integrability cubic Schr{\"o}dinger equation.