TheREUpresentation

# TheREUpresentation - Quantization Theorem 1 Fast Fourier...

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Quantization Theorem 1 Fast Fourier Transform Dispersive Quantization Sudarshan Balakrishnan 1 Gong Chen 2 1 Department of Mathematics University of Michigan 2 Department of Mathematics University of Minnesota Research Experience for Undergraduates, 2011 Sudarshan Balakrishnan, Gong Chen

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Quantization Theorem 1 Fast Fourier Transform Outline 1 Quantization Previous Work Deﬁnitions Linearly dispersive wave equation. Schrödinger wave equation. 2 Theorem 1 Proof 3 Fast Fourier Transform Verifying Solutions Numerical Error Sudarshan Balakrishnan, Gong Chen
Quantization Theorem 1 Fast Fourier Transform Previous Work Deﬁnitions Linearly dispersive wave equation. Schrödinger wave equation. Outline 1 Quantization Previous Work Deﬁnitions Linearly dispersive wave equation. Schrödinger wave equation. 2 Theorem 1 Proof 3 Fast Fourier Transform Verifying Solutions Numerical Error Sudarshan Balakrishnan, Gong Chen

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Quantization Theorem 1 Fast Fourier Transform Previous Work Deﬁnitions Linearly dispersive wave equation. Schrödinger wave equation. Previous Work Physicists and mathematicians have observed that the linear Schrödinger equation shows signiﬁcantly different behavior at rational and irrational times for special initial data. “ Does a quantum particle know time?” written by Lev Kapotanski and Igor Rodnianski provides interesting insights into the behavior of the Schrödinger equation . Sudarshan Balakrishnan, Gong Chen
Quantization Theorem 1 Fast Fourier Transform Previous Work Deﬁnitions Linearly dispersive wave equation. Schrödinger wave equation. Previous Work Olver proved similar results for linear KdV and pointed out his method works for all dispersive equations with dispersive relation is polynomial in Q [ x ] But Prof. Olver did not pay much attention to the phenomenon when time is irrational. He only said that when time is irrational the graph is fractal without proof. Sudarshan Balakrishnan, Gong Chen

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Quantization Theorem 1 Fast Fourier Transform Previous Work Deﬁnitions Linearly dispersive wave equation. Schrödinger wave equation. Previous Work Hardy used identities of Jacobi Ellipitic function to show the fundamental solution of linear Schrödinger wave equation is divergent everywhere. Oskolkov used Vinogradov’s method to show that when t is irrational all dispersive equation whose dispersive relation is polynomial in Q [ x ] with step function special initial data is continuous. So, when t is irrational the solution of the Schrödinger wave equation is nowhere differentiable.
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TheREUpresentation - Quantization Theorem 1 Fast Fourier...

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