文档介绍：The Journal of Fourier Analysis and Applications
Volume 4, Issue 3, 1998
Sehr6dinger Equation and
Oscillatory Hilbert Transforms
of Second Degree
K. Oskolkov
Communicated by Andrew M. Odlyzko
ABSTRACT. Let
e~r i (tn 2 +2xn) v"'s e~r i (tn 2 +2xn)
= lira ~
h(t,x) := . E 2~rin N~ 27tin
n~Z\{0} 0<lnl_<N
(i = .J-L-T; t, x - real variables). It is proved that in the rectangle D := { (t, x) : 0 < t < 1, Ix I < },
the function h satisfies the following functional inequality:
[h<t,x,[ < ~7 h(~,t) -Pc,
where c is an absolute positive constant. Iterations of this relation provide another, more elementary,
proof of the known global boundedness result
IIh: L~ (E2)[I := ess sup Ih<t,x)l < ~.
The above functional inequality is derived from a general duality relation, of them-function type, for
solutions of the Cauchy initial value problem for Schri~dinger equation of a free particle.
Variation plexity of solutions of Schri~dinger equation are discussea~
1. Schr6dinger Equation, Functional Relations and L~-Result
Consider the Cauchy initial value problem for time-dependent Schrtdinger equation of a free
particle
0up 1 02up
0t - 4zri 0x 2' ~(t, x)It=o f(x). ()
Math Subject Classifications. 42A16, 35J10, 11L07, 11T24, 33B20.
Keywords and Phrases. Cauchy initial value problem, Schrtdinger equation, Hilbert transform, Gauss' sum,
continued fraction, Fresnel integral, curlicue, selfsimilarity.
Acknowledgements and Notes. The author was supported by DEPSCOR Grant N000149611003 and NSF Grant
No. DMS 9706883. The author expresses his gratitude to Irina Mitrea, who read the manuscript and made a
number of valuable remarks.
1998 Birkhiiuser Boston. All rights reserved
ISSN 1069-5869
342 K. Oskolkov
~tix 2 7ri
The Green's function of this problem is F(t, x) = V/ff• e , , with ~r := e'a" = ~ and
~/7 := i~/~, t < 0; the solution operator ~p(f; t, x) is represented by the convolution