Algebras of commuting differential operators for integral kernels of Airy type

Autores
Casper, W. Riley; Grünbaum, Francisco Alberto; Yakimov, Milen; Zurrián, Ignacio Nahuel
Año de publicación
2021
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.
Fil: Casper, W. Riley. California State University, Fullerton; Estados Unidos
Fil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados Unidos
Fil: Yakimov, Milen. Northeastern University; Estados Unidos
Fil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Materia
Mathematical physics
Rings and algebras
Spectral theory
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/172708
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Algebras of commuting differential operators for integral kernels of Airy typeCasper, W. RileyGrünbaum, Francisco AlbertoYakimov, MilenZurrián, Ignacio NahuelMathematical physicsRings and algebrasSpectral theoryhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.Fil: Casper, W. Riley. California State University, Fullerton; Estados UnidosFil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados UnidosFil: Yakimov, Milen. Northeastern University; Estados UnidosFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaCornell University2021-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/172708Casper, W. Riley; Grünbaum, Francisco Alberto; Yakimov, Milen; Zurrián, Ignacio Nahuel; Algebras of commuting differential operators for integral kernels of Airy type; Cornell University; arXiv; 12-2021; 1-192331-8422CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.48550/arXiv.2112.11639info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/2112.11639info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:10:10Zoai:ri.conicet.gov.ar:11336/172708instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:10:10.772CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Algebras of commuting differential operators for integral kernels of Airy type
title Algebras of commuting differential operators for integral kernels of Airy type
spellingShingle Algebras of commuting differential operators for integral kernels of Airy type
Casper, W. Riley
Mathematical physics
Rings and algebras
Spectral theory
title_short Algebras of commuting differential operators for integral kernels of Airy type
title_full Algebras of commuting differential operators for integral kernels of Airy type
title_fullStr Algebras of commuting differential operators for integral kernels of Airy type
title_full_unstemmed Algebras of commuting differential operators for integral kernels of Airy type
title_sort Algebras of commuting differential operators for integral kernels of Airy type
dc.creator.none.fl_str_mv Casper, W. Riley
Grünbaum, Francisco Alberto
Yakimov, Milen
Zurrián, Ignacio Nahuel
author Casper, W. Riley
author_facet Casper, W. Riley
Grünbaum, Francisco Alberto
Yakimov, Milen
Zurrián, Ignacio Nahuel
author_role author
author2 Grünbaum, Francisco Alberto
Yakimov, Milen
Zurrián, Ignacio Nahuel
author2_role author
author
author
dc.subject.none.fl_str_mv Mathematical physics
Rings and algebras
Spectral theory
topic Mathematical physics
Rings and algebras
Spectral theory
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.
Fil: Casper, W. Riley. California State University, Fullerton; Estados Unidos
Fil: Grünbaum, Francisco Alberto. University of California at Berkeley; Estados Unidos
Fil: Yakimov, Milen. Northeastern University; Estados Unidos
Fil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
description Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux transformations of the Airy function in terms of the fixed point submanifold of a differential Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy Adelic Grassmannian and initiate the study of the full algebra of differential operators commuting with each of the integral operators in question. We leverage the general theory of [8] to obtain explicit formulas for the two differential operators of lowest orders that commute with each of the level one and two integral operators obtained in the Darboux process. Moreover, we prove that each pair of differential operators commute with each other. The commuting operators in the level one case are shown to satisfy an algebraic relation defining an elliptic curve.
publishDate 2021
dc.date.none.fl_str_mv 2021-12
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/172708
Casper, W. Riley; Grünbaum, Francisco Alberto; Yakimov, Milen; Zurrián, Ignacio Nahuel; Algebras of commuting differential operators for integral kernels of Airy type; Cornell University; arXiv; 12-2021; 1-19
2331-8422
CONICET Digital
CONICET
url http://hdl.handle.net/11336/172708
identifier_str_mv Casper, W. Riley; Grünbaum, Francisco Alberto; Yakimov, Milen; Zurrián, Ignacio Nahuel; Algebras of commuting differential operators for integral kernels of Airy type; Cornell University; arXiv; 12-2021; 1-19
2331-8422
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.48550/arXiv.2112.11639
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/2112.11639
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Cornell University
publisher.none.fl_str_mv Cornell University
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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