- Dispersive Quantization Peter J Olver School of Mathematics University of Minnesota Minneapolis MN 55455 [email protected]

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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 initial-boundary 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 fractal-like, non-diFerentiable 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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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 Talbot-like 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.

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- Dispersive Quantization Peter J Olver School of Mathematics University of Minnesota Minneapolis MN 55455 [email protected]

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