Full paper in PDF:
$%O. Lopes, Instability of radial standing waves of Schrödinger equation on the exterior of a ball, Rev. Mat. Complut. 12 (1999), no. 2, 537–545.%$

Instability of Radial Standing Waves of Schrödinger Equation on the Exterior of a Ball
Orlando LOPES
IMECC-UNICAMP
C.P. 6065
13083-970 Campinas-SP — Brazil
lopes@ime.unicamp.br

Received: November 11, 1998
Revised: February 4, 1999

ABSTRACT

Under smoothness and growth assumptions on f we show that a standing wave ibt w(t,x)= e f(x) of the Schrödinger equation on the exterior _O_ of a ball and Neumann boundary condition

2 @w- wt = i(Dw +f(|w|)w) @n =0 on @_O_

where b is real and f is real and radially symmetric, is always linearly unstable under perturbations in the space H1(_O_) . (It may be stable under perturbations in H1rad(_O_) .)

The instability is independent of f having a fixed sign and of its Morse index.

The main tool is a theorem of linearized instability of M. Grillakis.

1991 Mathematics Subject Classification: 35Q55.