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Dispersive Quantization
Peter J. Olver
†
School of Mathematics
University of Minnesota
Minneapolis, MN
55455
[email protected]
http://www.math.umn.edu/
∼
olver
Abstract.
The evolution, through linear dispersion, of piecewise constant periodic
initial data leads to surprising quantized structures at rational times, and fractal, nondiFer
entiable pro±les at irrational times. Similar phenomena have been observed in optics and
quantum mechanics, and lead to intriguing connections with exponential sums arising in
number theory. Rami±cations of these observations for numerics and nonlinear dispersion
are proposed as open problems.
1. Introduction.
The genesis of this note was a supposedly straightforward exercise, based on a simple
initialboundary value problem for linearly dispersive waves in a periodic domain, that I
had devised for my forthcoming text in partial diFerential equations, [
10
]. Constructing
the ²ourier series solution is not especially challenging, and so I also asked for graphs of
the solution at various times. In the course of writing up the solution, the initial plots
that I produced with
Mathematica
were more or less as predicted; however, when I
introduced a diFerent time step, the solution exhibited a completely unexpected behavior.
²urther experimentation revealed that the solution has a fractallike, nondiFerentiable
structure at irrational times, but is piecewise constant at rational times! I had never
seen anything like this before, but the fact that the problem was so elementary convinced
me that it must be well known. Nevertheless, all of the leading experts in dispersive
waves to whom I showed these computations were similarly surprised, convincing me that
†
Supported in part by NSF Grant DMS
08–07317
.
March 1, 2011
1
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View Full Document I might be onto something new and of potential signiFcance. Some further digging revealed
that such rational/irrational behavior had already been noted in the context of the linear
Schr¨
odinger equation, and leads rapidly into the deep waters of advanced ±ourier analysis
and exponential sums in number theory. This in turn pointed me towards a recent series
of papers by Michael Berry and his collaborators, [
1
,
3
,
4
], that describe the
Talbot efect
in optics and in quantum revivals, which was named after a striking optical experiment,
[
15
], conducted in 1836 by William Henry ±ox Talbot, one of the founders of photography.
Thus, the present note reveals that a similar Talbotlike quantized/fractal e²ect can also
be found in a broad range of dispersive media.
Although this paper only makes a modest mathematical contribution, the fact that
these phenomena are not well known has convinced me that it is worth setting down in
print. The paper contains more questions than results, and the implications for wave
mechanics, both linear and nonlinear, and numerics remain to be explored.
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This note was uploaded on 11/19/2011 for the course MATH 101 taught by Professor Wormer during the Spring '08 term at UCSD.
 Spring '08
 wormer
 Math

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