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Summary: Mathematical Research Letters , 1 ()
CONVEXITY PROPERTIES OF SOLUTIONS TO THE FREE
SCHRšODINGER EQUATION WITH GAUSSIAN DECAY
L. Escauriaza, C.E. Kenig, G. Ponce, and L. Vega
Abstract. We study convexity properties of solutions to the free Schršodinger
equation with Gaussian decay.
1. Introduction
The main purpose of this note is to study convexity properties of two classes
of functions. The first one is the class of functions which, together with their
Fourier transform, have Gaussian decay. The second class is the class of functions
for which the free Schršodinger evolution has Gaussian decay at two different
times. (We will see later that, in fact, the two classes coincide). As motivation
for the study of these issues, we recall the well known "uncertainty principle"
due to G. H. Hardy (see [9]) :
(A) Assume n = 1, f(x) = O(e-Ax2
) and ^f() = O(e-B2
). If AB > 1/4,
then f 0. Moreover, if AB = 1/4, then f(x) = ce-Ax2
, for some constant c.
In [8] this result was extended to higher dimensions, with x2
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