scholarly journals PERTURBATION OF THE SEMICLASSICAL SCHRÖDINGER EQUATION ON NEGATIVELY CURVED SURFACES

2015 ◽  
Vol 16 (4) ◽  
pp. 787-835 ◽  
Author(s):  
Suresh Eswarathasan ◽  
Gabriel Rivière
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Cotangent Bundle ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Quantum Evolution ◽  
Curved Surfaces ◽  
Small Perturbations ◽  
Negatively Curved ◽  
Ehrenfest Time

We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit, and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

scholarly journals WKB-Based Schemes for the Oscillatory 1D Schrödinger Equation in the Semiclassical Limit

2011 ◽  
Vol 49 (4) ◽  
pp. 1436-1460 ◽  
Author(s):  
Anton Arnold ◽  
Naoufel Ben Abdallah ◽  
Claudia Negulescu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

scholarly journals Finite speed of disturbance for the nonlinear Schrödinger equation

2018 ◽  
Vol 149 (6) ◽  
pp. 1405-1419
Author(s):  
Simão Correia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Whole Space

AbstractWe consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

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