Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents

Handle URI:
http://hdl.handle.net/10754/583035
Title:
Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents
Authors:
Athanassoulis, Agissilaos; Katsaounis, Theodoros ( 0000-0001-7387-7987 ) ; Kyza, Irene
Abstract:
Semiclassical asymptotics for Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P.L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM. © 2016 International Press.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Athanassoulis A, Katsaounis T, Kyza I (2016) Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents. Communications in Mathematical Sciences 14: 1821–1858. Available: http://dx.doi.org/10.4310/CMS.2016.v14.n7.a3.
Publisher:
International Press of Boston
Journal:
Communications in Mathematical Sciences
Issue Date:
30-Aug-2016
DOI:
10.4310/CMS.2016.v14.n7.a3
ARXIV:
arXiv:1403.7935
Type:
Article
ISSN:
1539-6746; 1945-0796
Sponsors:
Part of this work was completed while Th. Katsaounis was visiting the Dept. of Mathematics of Univ. of Leicester, UK. The author would like to thank prof. E. Georgoulis and the department for their hospitality and support.
Additional Links:
http://www.intlpress.com/site/pub/pages/journals/items/cms/content/vols/0014/0007/a003/; http://arxiv.org/abs/1403.7935; http://adsabs.harvard.edu/abs/2014arXiv1403.7935A
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorAthanassoulis, Agissilaosen
dc.contributor.authorKatsaounis, Theodorosen
dc.contributor.authorKyza, Ireneen
dc.date.accessioned2015-12-01T07:30:08Zen
dc.date.available2015-12-01T07:30:08Zen
dc.date.issued2016-08-30en
dc.identifier.citationAthanassoulis A, Katsaounis T, Kyza I (2016) Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents. Communications in Mathematical Sciences 14: 1821–1858. Available: http://dx.doi.org/10.4310/CMS.2016.v14.n7.a3.en
dc.identifier.issn1539-6746en
dc.identifier.issn1945-0796en
dc.identifier.doi10.4310/CMS.2016.v14.n7.a3en
dc.identifier.urihttp://hdl.handle.net/10754/583035en
dc.description.abstractSemiclassical asymptotics for Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P.L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM. © 2016 International Press.en
dc.description.sponsorshipPart of this work was completed while Th. Katsaounis was visiting the Dept. of Mathematics of Univ. of Leicester, UK. The author would like to thank prof. E. Georgoulis and the department for their hospitality and support.en
dc.language.isoenen
dc.publisherInternational Press of Bostonen
dc.relation.urlhttp://www.intlpress.com/site/pub/pages/journals/items/cms/content/vols/0014/0007/a003/en
dc.relation.urlhttp://arxiv.org/abs/1403.7935en
dc.relation.urlhttp://adsabs.harvard.edu/abs/2014arXiv1403.7935Aen
dc.rightsArchived with thanks to Communications in Mathematical Sciencesen
dc.subjectA posteriori error controlen
dc.subjectMultivalued flowen
dc.subjectSelection principleen
dc.subjectSemiclassical limit for rough potentialen
dc.subjectWigner transformen
dc.titleRegularized semiclassical limits: Linear flows with infinite Lyapunov exponentsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalCommunications in Mathematical Sciencesen
dc.eprint.versionPost-printen
dc.contributor.institutionDepartment of Mathematics, University of Leicester, United Kingdomen
dc.contributor.institutionIACM-FORTH, Heraklion, Greeceen
dc.contributor.institutionDivision of Mathematics, University of Dundee, Dundee, Scotland, United Kingdomen
dc.contributor.institutionInstitute of Applied and Computational Mathematics-FORTH, Nikolaou Plastira 100, Vassilika Vouton, Heraklion- Crete, Greeceen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
dc.identifier.arxividarXiv:1403.7935en
kaust.authorKatsaounis, Theodorosen
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