Maximal determinants of Schrödinger operators on bounded intervals
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 803-829.

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 L q -norm restriction (q1). This is done by first extending the definition of the functional determinant to the case of L q 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 q1, providing a complete characterisation of extremal potentials in the case where q 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 L q (q1). On commence par étendre la définition du déterminant fonctionnel au cas de potentiels L q , 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 q1, et les principales propriétés de ces solutions sont étudiées. On donne une caractérisation complète des potentiels optimaux dans les cas q=1 (fonction créneau) et q=2 (fonction de Weierstraß).

Received:
Accepted:
Published online:
DOI: 10.5802/jep.128
Classification: 11M36,  34L40,  49J15
Keywords: Functional determinant, extremal spectra, Pontryagin maximum principle, Weierstraß -function
Clara L. Aldana 1; Jean-Baptiste Caillau 2; Pedro Freitas 3

1 Universidad del Norte Vía Puerto Colombia, Barranquilla, Colombia
2 Université Côte d’Azur, CNRS, Inria, LJAD, France
3 Departamento de Matemática, Instituto Superior Técnico Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal and Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal
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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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