Ballistic Transport for Limit-periodic Schr\"odinger Operators in One Dimension

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Giorgio Young
February 9, 2023
11:30AM - 12:30PM
Location
MW 154

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Add to Calendar 2023-02-09 11:30:00 2023-02-09 12:30:00 Ballistic Transport for Limit-periodic Schr\"odinger Operators in One Dimension Title:  Ballistic Transport for Limit-periodic Schr\"odinger Operators in One Dimension Speaker:  Giorgio Young (University of Michigan) Speaker's URL:  https://lsa.umich.edu/math/people/postdoc-faculty/gfyoung.html Abstract:  In this talk, we consider the transport properties of the class of limit-periodic continuum Schr\"odinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator $H$, and $X_H(t)$ the Heisenberg evolution of the position operator, we show the limit of $\frac{1}{t}X_H(t)\psi$ as $t\to\infty$ exists and is nonzero for $\psi\ne 0$ belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time. URL associated with Seminar:  https://u.osu.edu/aots/ MW 154 Department of Mathematics math@osu.edu America/New_York public
Description

Title:  Ballistic Transport for Limit-periodic Schr\"odinger Operators in One Dimension

Speaker:  Giorgio Young (University of Michigan)

Speaker's URL:  https://lsa.umich.edu/math/people/postdoc-faculty/gfyoung.html

Abstract:  In this talk, we consider the transport properties of the class of limit-periodic continuum Schr\"odinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator $H$, and $X_H(t)$ the Heisenberg evolution of the position operator, we show the limit of $\frac{1}{t}X_H(t)\psi$ as $t\to\infty$ exists and is nonzero for $\psi\ne 0$ belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.

URL associated with Seminar:  https://u.osu.edu/aots/

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