We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions under an -norm restriction (). This is done by first extending the definition of the functional determinant to the case of potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all , providing a complete characterisation of extremal potentials in the case where is one (a pulse) and two (Weierstraß’s function).
On cherche les potentiels qui maximisent le déterminant fonctionnel d’un opérateur de Schrödinger sur un intervalle borné, avec conditions aux limites de Dirichlet et sous contrainte de norme (). On commence par étendre la définition du déterminant fonctionnel au cas de potentiels , en montrant que le problème de maximisation associé est équivalent à un problème de contrôle optimal. On prouve l’existence et l’unicité de solution de ce problème pour tout , et les principales propriétés de ces solutions sont étudiées. On donne une caractérisation complète des potentiels optimaux dans les cas (fonction créneau) et (fonction de Weierstraß).
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Keywords: Functional determinant, extremal spectra, Pontryagin maximum principle, Weierstraß $\wp $-function
Mots-clés : Déterminant fonctionnel, extrema spectraux, principe du maximum de Pontrjagin, fonction $\wp $ de Weierstraß
Clara L. Aldana 1; Jean-Baptiste Caillau 2; Pedro Freitas 3
CC-BY 4.0
@article{JEP_2020__7__803_0,
author = {Clara L. Aldana and Jean-Baptiste Caillau and Pedro Freitas},
title = {Maximal determinants of {Schr\"odinger~operators} on bounded intervals},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {803--829},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
doi = {10.5802/jep.128},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.128/}
}
TY - JOUR AU - Clara L. Aldana AU - Jean-Baptiste Caillau AU - Pedro Freitas TI - Maximal determinants of Schrödinger operators on bounded intervals JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 803 EP - 829 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.128/ DO - 10.5802/jep.128 LA - en ID - JEP_2020__7__803_0 ER -
%0 Journal Article %A Clara L. Aldana %A Jean-Baptiste Caillau %A Pedro Freitas %T Maximal determinants of Schrödinger operators on bounded intervals %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 803-829 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.128/ %R 10.5802/jep.128 %G en %F JEP_2020__7__803_0
Clara L. Aldana; Jean-Baptiste Caillau; Pedro Freitas. Maximal determinants of Schrödinger operators on bounded intervals. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 803-829. doi : 10.5802/jep.128. https://jep.centre-mersenne.org/articles/10.5802/jep.128/
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