Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations

Katherine A. Kime

doi:10.3934/dcdsb.2018063

Article Contents

2018, Volume 23, Issue 4: 1601-1621. Doi: 10.3934/dcdsb.2018063

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Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations

Katherine A. Kime^,

Department of Mathematics and Statistics, University of Nebraska Kearney, Kearney, Nebraska 68849, USA

* Corresponding author: Katherine A. Kime
* Corresponding author: Katherine A. Kime

Received: May 2017

Early access: February 2018

Published: April 2018

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Abstract

We consider discrete potentials as controls in systems of finite difference equations which are discretizations of a 1-D Schrödinger equation. We give examples of palindromic potentials which have corresponding steerable initial-terminal pairs which are not mirror-symmetric. For a set of palindromic potentials, we show that the corresponding steerable pairs that satisfy a localization property are mirror-symmetric. We express the initial and terminal states in these pairs explicitly as scalar multiples of vector-valued functions of a parameter in the control.

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Mathematics Subject Classification: Primary: 93B03, 93B40, 93C20; Secondary: 81Q05, 81Q93.

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Figure 1. Example 1. $\alpha$-Localized, Mirror-Symmetric

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Figure 2. Example 2. Not Localized, Not Mirror-Symmetric

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Figure 3. Example 3. Localized with Equal Degree of Restriction Equal to 1, Not $\alpha$-Localized, Not Mirror-Symmetric

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