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

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1601-1621
Number of pages21
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume23
Issue number4
DOIs
StatePublished - Jun 2018

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Mirror Symmetry
Finite Difference
Mirrors
Discretization
Mirror
Difference equations
Finite Difference Equation
Vector-valued Functions
Express
Scalar

Keywords

  • Complex-valued matrix
  • Control
  • Mirror
  • Palindromic
  • Potential
  • Schrödinger
  • Symmetry

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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abstract = "We consider discrete potentials as controls in systems of finite difference equations which are discretizations of a 1-D Schr{\"o}dinger 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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AB - 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.

KW - Complex-valued matrix

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KW - Potential

KW - Schrödinger

KW - Symmetry

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