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Series solutions of the non-stationary Heun equation
KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.ORCID iD: 0000-0003-1839-8128
KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the non-stationary Heun equation, also known as quantum PainlevéVI, which has appeared in dierent works on quantum integrable models and conformaleld theory. We use a generalized kernel function identity to transform the problemto solve this equation into a dierential-dierence equation which, as we show, canbe solved by ecient recursive algorithms. We thus obtain series representations ofsolutions which provide elliptic generalizations of the Jacobi polynomials. These seriesreproduces, in a limiting case, a perturbative solution of the Heun equation due toTakemura, but our method is dierent in that we expand in non-conventional basisfunctions that allow us to obtain explicit formulas to all orders;

Keyword [en]
Heun equation, Lamé equation, Kernel functions, quantum Painlevé VI, perturbation theory
National Category
Other Physics Topics
Identifiers
URN: urn:nbn:se:kth:diva-193357OAI: oai:DiVA.org:kth-193357DiVA: diva2:1010347
Note

QC 20161004

Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2016-10-04Bibliographically approved
In thesis
1. A kernel function approach to exact solutions of Calogero-Moser-Sutherland type models
Open this publication in new window or tab >>A kernel function approach to exact solutions of Calogero-Moser-Sutherland type models
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This Doctoral thesis gives an introduction to the concept of kernel functionsand their signicance in the theory of special functions. Of particularinterest is the use of kernel function methods for constructing exact solutionsof Schrodinger type equations, in one spatial dimension, with interactions governedby elliptic functions. The method is applicable to a large class of exactlysolvable systems of Calogero-Moser-Sutherland type, as well as integrable generalizationsthereof. It is known that the Schrodinger operators with ellipticpotentials have special limiting cases with exact eigenfunctions given by orthogonalpolynomials. These special cases are discussed in greater detail inorder to explain the kernel function methods with particular focus on the Jacobipolynomials and Jack polynomials.

Place, publisher, year, edition, pages
Stockholm: Kungliga Tekniska högskolan, 2016. 57 p.
Series
TRITA-FYS, ISSN 0280-316X ; 2016:58
Keyword
Kernel functions, Calogero-Moser-Sutherland models, Ruijsenaarsvan Diejen models, Elliptic functions, Exact solutions, Source Identities, Chalykh- Feigin-Sergeev-Veselov type deformations, non-stationary Heun equation
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Other Physics Topics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-193322 (URN)978-91-7729-132-9 (ISBN)
Public defence
2016-10-27, Oskar Kleins auditorium FR4, Roslagstullsbacken 21, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20161003

Available from: 2016-10-04 Created: 2016-09-30 Last updated: 2016-10-04Bibliographically approved

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