Display Abstract

Title Exponential decay and resonances for Systems of Schrodinger equations

Name Claudio A Fernandez
Country Chile
Email cfernand@mat.puc.cl
Co-Author(s) M.A.Astaburuaga (PUC Chile), R. Coimbra Charao (Santa Catarina, Brasil) and G.Perla Menzala (LNCC, UFRJ, Brasil)
Submit Time 2014-02-26 08:41:38
Special Session 60: Recent advances in evolutionary equations
Let $k$ be a resonance (scattering frequency) for the operator $H =-\Delta \,+ \,V(x)$. This means that $\Im k$ is negative and the problem $$ (-\Delta \,+ \,V(x)\,) \varphi \,= k^2 \varphi, $$ has a nontrivial outgoing solution. Since the outgoing condition rules out square integrability, it is reasonable to truncate the resonant solution $\varphi$ in a compact region. One then expects that the evolution of the truncated solution under the Schrodinger equation, will exhibit an exponential behavior, when the time $t$ approaches infinity. We shall review a method, due to R. Lavine, that proves this fact on the half line $[0,\infty)$ and then extend it to systems of Schr\"odinger equations, $$-\Delta u + V(x)u + g(x)v \,=k^2 u \, , $$ $$ -\Delta v + W(x)v + g(x)u \, =k^2 v \, . $$