Maximal determinants of Schrödinger operators on bounded intervals

Aldana, Clara L.; Caillau, Jean-Baptiste; Freitas, Pedro

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 803-829.

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 $L^q$ ($q\ge 1$). 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 $q\ge 1$, 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 $\wp$ de Weierstraß).

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 ($q\ge 1$). 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 $q\ge 1$, providing a complete characterisation of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstraß’s $\wp$ function).

Classification : 11M36, 34L40, 49J15
Keywords: Functional determinant, extremal spectra, Pontryagin maximum principle, Weierstraß $\wp $-function
Mot clés : Déterminant fonctionnel, extrema spectraux, principe du maximum de Pontrjagin, fonction $\wp $ de Weierstraß

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