Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations

Abstract

The asymptotic behavior of solutions to the Cauchy problem for the equation

$$i\psi _\imath = \frac{1}{2}\Delta \psi - \upsilon (\psi )\psi , \upsilon = r^{ - 1} *|\psi |^2 ,$$

and for systems of similar form, is studied. It is shown that the norms

$$\parallel \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 + \parallel \nabla \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 $$

are integrable in time for any fixedR>0, from which it follows that

$$\mathop {\lim }\limits_{t \to \infty } \parallel \psi (t)\parallel _{L_2 (|x| \leqq R)} = 0.$$

\] Nevertheless, it is established that anL ₂-scattering theory is impossible.