Metric cycles, curves and solenoids
Vladimir Georgiev Eugene Stepanov
We prove that every one-dimensional real Ambrosio-Kirchheim current with zero boundary (i.e. a cycle) in a lot of reasonable spaces (including all finite-dimensional normed spaces) can be represented by a Lipschitz curve parameterized over the real line through a suitable limit of Cesàro means of this curve over a subsequence of symmetric bounded intervals (viewed as currents). It is further shown that in such spaces, if a cycle is indecomposable, i.e. does not contain ``nontrivial'' subcycles, then it can be represented again by a Lipschitz curve parameterized over the real line through a limit of Cesàro means of this curve over every sequence of symmetric bounded intervals, that is, in other words, such a cycle is a solenoid.
keywords: Ambrosio-Kirchheim currents solenoids. metric currents Lipschitz curves Normal currents
Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one
Daniele Garrisi Vladimir Georgiev

We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

keywords: Stability uniqueness Schrödinger
Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential
Vladimir Georgiev Atanas Stefanov Mirko Tarulli
The work treats smoothing and dispersive properties of solutions to the Schrödinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing - Strichartz estimate for the corresponding Cauchy problem. An application that guarantees absence of pure point spectrum of the corresponding perturbed Laplace operator is discussed too.
keywords: Strichartz estimates smoothing properties. Schrödinger Equation

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