Maximal determinants of Schrödinger operators on bounded intervals

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

doi:10.5802/jep.128

Maximal determinants of Schrödinger operators on bounded intervals
[Maximisation du déterminant d’opérateurs de Schrödinger sur un intervalle borné]

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

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

Résumé
Abstract

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 $L^{q}$ ( $q \geq 1$ ). 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 $q \geq 1$ , 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 $℘$ 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 ( $q \geq 1$ ). 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 $q \geq 1$ , providing a complete characterisation of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstraß’s $℘$ function).

Reçu le : 2019-08-17
Accepté le : 2020-04-28
Publié le : 2020-05-18

DOI : 10.5802/jep.128

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

Affiliations des auteurs :

Clara L. Aldana ¹ ; Jean-Baptiste Caillau ² ; Pedro Freitas ³

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Maximal determinants of {Schr\"odinger~operators} on bounded intervals},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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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, Tome 7 (2020), pp. 803-829. doi : 10.5802/jep.128. https://jep.centre-mersenne.org/articles/10.5802/jep.128/

Bibliographie
Cité par

[1] A. A. Agrachev & Y. L. Sachkov - Control theory from the geometric viewpoint, Encyclopaedia of Math. Sciences, vol. 87, Springer-Verlag, Berlin, 2004 | DOI | MR | Zbl

[2] P. Albin, C. L. Aldana & F. Rochon - “Ricci flow and the determinant of the Laplacian on non-compact surfaces”, Comm. Partial Differential Equations 38 (2013) no. 4, p. 711-749 | DOI | MR | Zbl

[3] E. Aurell & P. Salomonson - “On functional determinants of Laplacians in polygons and simplicial complexes”, Comm. Math. Phys. 165 (1994) no. 2, p. 233-259 | DOI | MR | Zbl

[4] B. Bonnard & M. Chyba - Singular trajectories and their role in control theory, Math. & Applications, vol. 40, Springer-Verlag, Berlin, 2003 | MR | Zbl

[5] D. Burghelea, L. Friedlander & T. Kappeler - “On the determinant of elliptic boundary value problems on a line segment”, Proc. Amer. Math. Soc. 123 (1995) no. 10, p. 3027-3038 | DOI | MR | Zbl

[6] G. Buttazzo, A. Gerolin, B. Ruffini & B. Velichkov - “Optimal potentials for Schrödinger operators”, J. Éc. polytech. Math. 1 (2014), p. 71-100 | DOI | Numdam | Zbl

[7] L. Cesari - Optimization—theory and applications. Problems with ordinary differential equations, Applications of Math., vol. 17, Springer-Verlag, New York, 1983 | DOI | Zbl

[8] E. A. Coddington & N. Levinson - Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955 | Zbl

[9] H. M. Edwards - Riemann’s zeta function, Dover Publications, Inc., Mineola, NY, 2001 | MR | Zbl

[10] P. Freitas - “The spectral determinant of the isotropic quantum harmonic oscillator in arbitrary dimensions”, Math. Ann. 372 (2018) no. 3-4, p. 1081-1101 | DOI | MR | Zbl

[11] P. Freitas & J. Lipovsky - “The determinant of one-dimensional polyharmonic operators of arbitrary order”, 2020 | arXiv

[12] I. M. Gelfand & A. M. Jaglom - “Integration in functional spaces and its applications in quantum physics”, J. Math. Phys. 1 (1960), p. 48-69 | DOI | MR

[13] E. M. Harrell - “Hamiltonian operators with maximal eigenvalues”, J. Math. Phys. 25 (1984) no. 1, p. 48-51, Erratum: Ibid. 27 (1986), no. 1, p. 419 | DOI | MR | Zbl

[14] A. Henrot - Extremum problems for eigenvalues of elliptic operators, Frontiers in Math., Birkhäuser Verlag, Basel, 2006 | Zbl

[15] T. Kato - Perturbation theory for linear operators, Classics in Math., Springer-Verlag, Berlin, 1995 | Zbl

[16] M. Lesch - “Determinants of regular singular Sturm-Liouville operators”, Math. Nachr. 194 (1998), p. 139-170 | DOI | MR | Zbl

[17] M. Lesch & J. Tolksdorf - “On the determinant of one-dimensional elliptic boundary value problems”, Comm. Math. Phys. 193 (1998) no. 3, p. 643-660 | DOI | MR | Zbl

[18] S. Levit & U. Smilansky - “A theorem on infinite products of eigenvalues of Sturm-Liouville type operators”, Proc. Amer. Math. Soc. 65 (1977) no. 2, p. 299-302 | DOI | MR | Zbl

[19] S. Minakshisundaram & Å. Pleijel - “Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds”, Canad. J. Math. 1 (1949), p. 242-256 | DOI | MR | Zbl

[20] K. Okikiolu - “Critical metrics for the determinant of the Laplacian in odd dimensions”, Ann. of Math. (2) 153 (2001) no. 2, p. 471-531 | DOI | MR | Zbl

[21] B. Osgood, R. Phillips & P. Sarnak - “Extremals of determinants of Laplacians”, J. Funct. Anal. 80 (1988) no. 1, p. 148-211 | DOI | MR | Zbl

[22] D. B. Ray & I. M. Singer - “ $R$ -torsion and the Laplacian on Riemannian manifolds”, Adv. Math. 7 (1971), p. 145-210 | DOI | MR | Zbl

[23] B. Riemann - “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”, Monatsber. Berlin. Akad. (1859), p. 671-680, English translation in [9]

[24] A. M. Savchuk - “On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential”, Mat. Zametki 69 (2001) no. 2, p. 277-285 | DOI | MR | Zbl

[25] A. M. Savchuk & A. A. Shkalikov - “The trace formula for Sturm-Liouville operators with singular potentials”, Mat. Zametki 69 (2001) no. 3, p. 427-442 | DOI | MR | Zbl

[26] E. C. Titchmarsh - The theory of the Riemann zeta-function, The Clarendon Press, Oxford University Press, New York, 1986, Ed. by D. R. Heath-Brown | Zbl

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