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Summary: UNCERTAINTY PRINCIPLE OF MORGAN TYPE AND
SCHRšODINGER EVOLUTIONS
L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA
Abstract. We prove unique continuation properties for solutions of evolu-
tion Schršodinger equation with time dependent potentials. In the case of the
free solution these correspond to uncertainly principles referred to as being of
Morgan type. As an application of our method we also obtain results concern-
ing the possible concentration profiles of solutions of semi-linear Schršodinger
equations.
1. Introduction
In this paper we continue our study initiated in [5] [6], and [7] on unique contin-
uation properties of solutions of Schršodinger equations of the form
(1.1) itu + u = V (x, t)u, (x, t) Rn
Ś [0, 1].
The goal is to obtain sufficient conditions on the behavior of the solution u at
two different times and on the potential V which guarantee that u 0 in Rn
Ś[0, 1].
Under appropriate assumptions this result will allow us to extend these conditions
to the difference v = u1 - u2 of two solutions u1, u2 of semi-linear Schršodinger
equation
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