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A kernel function approach to exact solutions of Calogero-Moser-Sutherland type models
KTH, School of Engineering Sciences (SCI), Theoretical Physics.ORCID iD: 0000-0003-1839-8128
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 [en]
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
National Category
Other Physics Topics
Research subject
Physics
Identifiers
URN: urn:nbn:se:kth:diva-193322ISBN: 978-91-7729-132-9 (print)OAI: oai:DiVA.org:kth-193322DiVA: diva2:1001971
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
List of papers
1. Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models
Open this publication in new window or tab >>Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models
2014 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 104, no 7, 811-835 p.Article in journal (Refereed) Published
Abstract [en]

We consider the relativistic generalization of the quantum A (N-1) Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh-Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.

Keyword
exactly solvable models, Ruijsenaars models, Chalykh-Feigin-Veselov-Sergeev type deformation, kernel functions
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-147020 (URN)10.1007/s11005-014-0690-5 (DOI)000336412300002 ()2-s2.0-84901603241 (Scopus ID)
Funder
Swedish Research Council, 621-2010-3708
Note

QC 20140625

Available from: 2014-06-25 Created: 2014-06-23 Last updated: 2017-12-05Bibliographically approved
2. Deformed Calogero-Sutherland model and fractional Quantum Hall effect
Open this publication in new window or tab >>Deformed Calogero-Sutherland model and fractional Quantum Hall effect
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The deformed Calogero-Sutherland (CS) model is a quantum integrable systemwith arbitrary numbers of two types of particles and reducing to the standard CSmodel in special cases. We show that a known collective field description of theCS model, which is based on conformal field theory (CFT), is actually a collectivefield description of the deformed CS model. This provides a natural application ofthe deformed CS model in Wen’s effective field theory of the fractional quantumHall effect (FQHE), with the two kinds of particles corresponding to electrons andquasi-hole excitations. In particular, we use known mathematical results aboutsuper Jack polynomials to obtain simple explicit formulas for the orthonormal CFTbasis proposed by van Elburg and Schoutens in the context of the FQHE.

National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:kth:diva-193356 (URN)
Note

QC 20161004

Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2016-10-04Bibliographically approved
3. Series solutions of the non-stationary Heun equation
Open this publication in new window or tab >>Series solutions of the non-stationary Heun equation
(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
Heun equation, Lamé equation, Kernel functions, quantum Painlevé VI, perturbation theory
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-193357 (URN)
Note

QC 20161004

Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2016-10-04Bibliographically approved
4. Integral representation of solution to the non-stationary Lamé equation
Open this publication in new window or tab >>Integral representation of solution to the non-stationary Lamé equation
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider methods for constructing explicit solutions of the non-stationary Lame equation,which is a generalization of the classical Lame equation, that has appeared in works on integrablemodels, conformal eld theory, high energy physics and representation theory. We also present ageneral method for constructing integral representations of solutions to the non-stationary Lameequation by a recursive scheme. Explicit integral representations, for special values of the modelparameters, are also presented. Our approach is based on kernel function methods which can benaturally generalized to the non-stationary Heun equation.

Keyword
non-stationary Lame equation, kernel functions, solutions method, iterative integral representations
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-193358 (URN)
Note

QC 20161004

Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2016-10-04Bibliographically approved

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