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

  • * Corresponding author: Katherine A. Kime

    * Corresponding author: Katherine A. Kime
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  • 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.

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


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

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

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

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