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Summary: UNIQUE CONTINUATION FOR SCHRšODINGER EVOLUTIONS,
WITH APPLICATIONS TO PROFILES OF CONCENTRATION
AND TRAVELING WAVES
L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA
Abstract. We prove unique continuation properties for solutions of the evolu-
tion Schršodinger equation with time dependent potentials. As an application
of our method we also obtain results concerning the possible concentration
profiles of blow up solutions and the possible profiles of the traveling waves
solutions of semi-linear Schršodinger equations.
1. Introduction
In this paper we continue our study initiated in [8], [9], [10], and [11] on unique
continuation properties of solutions of Schršodinger equations. To begin with we
consider the linear equation
(1.1) tu = i(u + V (x, t)u), (x, t) Rn
Ś [0, ).
We shall be interested in finding the strongest possible space decay of global
solutions of (1.1). In this direction our first results are the following ones:
Theorem 1. Let u C([0, ) : L2
(Rn
)) be a solution of the equation (1.1) with a
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