In this paper we establish the equivalence of solutions between Schrödinger maps into S2 or H2 and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into H2 in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.

We consider the Schrödinger equation with derivative perturbation terms in one space dimension. For the linear equation, we show that the standard Strichartz estimates hold under specific smallness requirements on the potential. As an application, we establish existence of local solutions for quadratic derivative Schrödinger equations in one space dimension with small and rough Cauchy data.

A well known result of Jaffard states that an arbitrary region on a torus controls , in the L 2 sense, solutions of the free stationary and dynamical Schrödinger equations. In this note we show that the same result is valid in the presence of a potential, that is for Schrödinger operators, −∆ + V , V ∈ C ∞ .

Carbon nanotube field-effect transistors (CNTFETs) have been studied in recent years as a potential alternative to CMOS devices, because of the capability of ballistic transport. In order to account for the ballistic transport we solved the coupled Poisson and Schrödinger equations for the analysis these devices. Conventionally the coupled Schrödinger-Poisson equation is solved iteratively, by ...

We prove unique continuation properties for solutions of the evolution Schrödinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi-linear Schrödinger equations.

The purpose of this Comment is to show that the solutions to the zero energy Schrödinger equations for monomial central potentials discussed in a recently published Letter, may also be obtained from the corresponding free particle solutions in a straight forwardly way, using an algorithm previously devised by us. New solutions to the zero energy Schrödinger equation are also exhibited.

We survey recent wellposedness and illposedness results for the Cauchy problem for nonlinear Schrödinger equations in inhomogeneous media such as Riemannian manifolds or domains of the Euclidean space, trying to emphasize the influence of the geometry. The main tools are multilinear Strichartz estimates for the Schrödinger group. Mathematics Subject Classification (2000). Primary 35Q55; Seconda...

Singularly perturbed vector nonlinear Schrödinger equations (PVNLS) are investigated. Emphasis is placed upon the relation with their restriction: The singularly perturbed scalar nonlinear Schrödinger equation (PNLS) studied in [10]. It turns out that the persistent homoclinic orbit for the PNLS [10] is the only one for the PVNLS, asymptotic to the same saddle.

Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schrödinger equations depending on two variables and of nonlinear wave equations depending on three variables.