# School of Mathematics

## Prof Simon Ruijsenaars

Professor of Mathematical Physics

Applied Mathematics

### Contact details

Room: 9.14

Tel: +44 (0)113 3435182

Email: S.Ruijsenaars @ leeds.ac.uk

### Keywords

Mathematical physics

Integrable systems

Quantum soliton systems

Analytic difference operators

Special functions

## Research interests

My research interests belong to the area of finite-dimensional integrable systems. Most of my work is focussed on N-body quantum systems of soliton type. In such systems, momenta and bindings are conserved under scattering, and the scattering is factorised as if a sequence of 2-body collisions were taking place. The pertinent quantum systems and their classical versions are at the crossroads of a great many subfields of mathematics and physics. Their nonrelativistic versions date back to the mid-seventies and bear the names of Calogero, Moser and Sutherland. In the mid-eighties, relativistic versions of these systems were discovered by myself and my then student H. Schneider at the classical level. Shortly thereafter, I found a way to quantise the systems such that their integrable character is preserved, and I introduced relativistic versions of the classical and quantum Toda systems as well.

The first step in `solving' the relativistic systems consists in constructing/discovering joint eigenfunctions for the commuting Hamiltonians, which are analytic difference operators. The second step is to use these eigenfunctions to construct unitary Hilbert space transforms. (To be sure, neither the existence of joint eigenfunctions nor the feasibility to promote them to the kernel of a unitary transform follows from any known general results.) The sought-for transforms generalise Fourier transforms for the open Toda case and the hyperbolic version of the CMS systems, and Fourier series for the closed Toda case and the elliptic version. (The trigonometric version of the relativistic CMS systems has been solved in terms of the well-known Macdonald polynomials.)

The operators at issue have properties not shared by the well-studied classes of differential and discrete difference operators. Indeed, their symbols are exponential in the momenta, so that they do not even belong to the vast class of Fourier integral operators. As a consequence, a lot of novel problems arise in the construction of rigorous Hilbert space counterparts of the commuting analytic difference operators.

### Useful links

## Current postgraduate students

Steven Haworth (2012)

## Publications

**Haworth S, Ruijsenaars S** Hilbert space theory for relativistic dynamics with reflection. Special cases *Journal of Integrable Systems*, **1**, 2016

DOI:10.1093/integr/xyw003

View abstract

**Ruijsenaars SNM** Hilbert-Schmidt operators vs. integrable systems of elliptic calogero-moser type IV. The relativistic heun (van Diejen) case *Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)*, **11**, 2015

DOI:10.3842/SIGMA.2015.004

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**Hallnäs M, Ruijsenaars S** A recursive construction of joint eigenfunctions for the hyperbolic nonrelativistic calogero-moser Hamiltonians *International Mathematics Research Notices*, **2015**, 10278-10313

DOI:10.1093/imrn/rnu267

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**Hallnäs M, Ruijsenaars S** Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. I. First steps *Int. Math. Res. Notices*, 2013

DOI:10.1093/imrn/rnt076

**Ruijsenaars S** On Positive Hilbert-Schmidt Operators *Integral Equations and Operator Theory*, **75**, 393-407, 2013

DOI:10.1007/s00020-013-2034-8

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**Rains E, Ruijsenaars S** Difference operators of Sklyanin and van Diejen type *Communications in Mathematical Physics*, **320**, 851-889, 2013

DOI:10.1007/s00220-013-1692-3

**Feher L, Klimcik C, Ruijsenaars S** A NOTE ON THE GAUSS DECOMPOSITION OF THE ELLIPTIC CAUCHY MATRIX *J NONLINEAR MATH PHY*, **18**, 179-182, 2011

DOI:10.1142/S1402925111001477

**Ruijsenaars S** A Relativistic Conical Function and its Whittaker Limits *SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS*, **7**, 2011

DOI:10.3842/SIGMA.2011.101

**Nimmo JJC, Ruijsenaars SNM** Tzitzeica solitons vs. relativistic Calogero-Moser three-body clusters *J MATH PHYS*, **50**, 2009

DOI:10.1063/1.3110012

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**Ruijsenaars SNM** Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities *COMMUN MATH PHYS*, **286**, 629-657, 2009

DOI:10.1007/s00220-008-0707-y

**Ruijsenaars SNM** Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type. II. The A (N-1) Case: First Steps *COMMUN MATH PHYS*, **286**, 659-680, 2009

DOI:10.1007/s00220-008-0708-x

**Ruijsenaars SNM ** *Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case*SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 049 2009

DOI:10.3842/SIGMA.2009.049

**Ruijsenaars S ** Ruijsenaars-Schneider model. *This is an invited contribution to Scholarpedia*, 2009

**Ruijsenaars SNM** Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type III. The Heun case *Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)*, **5**, 2009

DOI:10.3842/SIGMA.2009.049

View abstract

**Ruijsenaars SNM** The classical hyperbolic Askey-Wilson dynamics without bound states *THEOR MATH PHYS+*, **154**, 418-432, 2008

**Ruijsenaars S, Stodolsky L** On the Euclidean version of the photon number integral *J MATH PHYS*, **49**, 2008

DOI:10.1063/1.2836411

**Ruijsenaars SNM ** *Quadratic transformations for a function that generalizes $_2F_1$ and the Askey-Wilson polynomials*RAMANUJAN JOURNAL, 339-364 2007

DOI:10.1007/s11139-006-0257-x

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**Chiang YM, Ruijsenaars SNM** On the Nevanlinna order of meromorphic solutions to linear analytic difference equations *STUD APPL MATH*, **116**, 257-287, 2006

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**Ruijsenaars SNM** Zero-eigenvalue eigenfunctions for differences of elliptic relativistic Calogero-Moser hamiltonians *THEOR MATH PHYS+*, **146**, 25-33, 2006

**Ruijsenaars SNM ** *A unitary joint eigenfunction transform for the A $\Delta$ Os $\exp(ia_{\pm} d/dz)+\exp(2 \pi z/a_{\mp})$*JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, 253-294 2005

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**Ruijsenaars SNM** The Hilbert space asymptotics of a class of orthonormal polynomials on a bounded interval *Developments in Mathematics*, **13**, 367-381, 2005

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**Ruijsenaars SNM ** *A relativistic hypergeometric function*JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 393-417 2005

DOI:10.1016/j.cam.2004.05.024

**Ruijsenaars S** Isometric reflectionless eigenfunction transforms for higher-order A$\Delta$Os *Journal of Nonlinear Mathematical Physics*, **12**, 565-598, 2005

**Friedman E, Ruijsenaars S** Shintani-Barnes zeta and gamma functions *ADV MATH*, **187**, 362-395, 2004

DOI:10.1016/j.aim.2003.07.020

**Ruijsenaars SNM ** *Parameter shifts, D-4 symmetry and joint eigenfunctions for commuting Askey-Wilson-type difference operators*JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 481-495 2004

DOI:10.1088/0305-4470/37/2/016

**Ruijsenaars SNM ** *Integrable BCN analytic difference operators: Hidden parameter symmetries and Eigenfunctions*NEW TRENDS IN INTEGRABILITY AND PARTIAL SOLVABILITY, 217-261 2004

**Ruijsenaars S** Relativistic Lamé functions; Completeness vs. polynomial asymptotics. *Indagationes Mathematicae*, **14**, 515-544, 2003

DOI:10.1016/S0019-3577(03)90059-0

**Ruijsenaars SNM** A generalized hypergeometric function II. Asymptotics and D-4 symmetry *COMMUN MATH PHYS*, **243**, 389-412, 2003

DOI:10.1007/s00220-003-0969-3

**Ruijsenaars S** A generalized hypergeometric function III. Associated Hilbert space transform. *Communications in Mathematical Physics*, **243**, 413-448, 2003

DOI:10.1007/s00220-003-0970-x

**Ruijsenaars S** Factorized weight functions vs. factorized scattering *Communications in Mathematical Physics*, **228**, 467-494, 2002

DOI:10.1007/s002200200662

**Ruijsenaars SNM** Reflectionless analytic difference operators III. Hilbert space aspects *J NONLINEAR MATH PHY*, **9**, 181-209, 2002

**Ruijsenaars S** A nonlocal Kac-van Moerbeke equation admitting N-soliton solutions *J NONLINEAR MATH PHY*, **9**, 192-206, 2002

**Ruijsenaars S** A new class of reflectionless second-order A$\Delta$Os and its relation to nonlocal solitons *Regular and chaotic dynamics*, **7**, 351-391, 2002

**Ruijsenaars SNM ** *Relativistic Lame functions revisited*JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 10595-10612 2001

**Degasperis A, Ruijsenaars SNM** Newton equivalent Hamiltonians for the harmonic oscillator *ANN PHYS-NEW YORK*, **293**, 92-109, 2001

**Ruijsenaars SNM ** *Self-adjoint A Delta Os with vanishing reflection*THEORETICAL AND MATHEMATICAL PHYSICS, 933-945 2001

**Ruijsenaars SNM** Reflectionless analytic difference operators II. Relations to soliton systems *J NONLINEAR MATH PHY*, **8**, 256-287, 2001

**Ruijsenaars SNM ** *Reflectionless analytic difference operators (A Delta Os): Examples, open questions and conjectures*JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, 240-248 2001

**Ruijsenaars SNM** Reflectionless analytic difference operators - I. Algebraic framework *J NONLINEAR MATH PHY*, **8**, 106-138, 2001

**Ruijsenaars SNM ** *Special functions defined by analytic difference equations*SPECIAL FUNCTIONS 2000: CURRENT PERSPECTIVE AND FUTURE DIRECTIONS, 281-333 2001

**Ruijsenaars SNM ** *Sine-Gordon solitons vs. relativistic Calogero-Moser particles*INTEGRABLE STRUCTURES OF EXACTLY SOLVABLE TWO-DIMENSIONAL MODELS OF QUANTUM FIELD THEORY, 273-292 2001

**Ruijsenaars SNM** Hilbert space theory for reflectionless relativistic potentials *PUBL RES I MATH SCI*, **36**, 707-753, 2000

**Ruijsenaars SNM** On Barnes' multiple zeta and gamma functions *ADV MATH*, **156**, 107-132, 2000

**Ruijsenaars SNM** A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type *COMMUN MATH PHYS*, **206**, 639-690, 1999

**Ruijsenaars SNM** Relativistic Lame functions: the special case g = 2 *J PHYS A-MATH GEN*, **32**, 1737-1772, 1999

**Ruijsenaars SNM** Generalized Lame functions. I. The elliptic case *J MATH PHYS*, **40**, 1595-1626, 1999

**Ruijsenaars SNM** Generalized Lame functions. II. Hyperbolic and trigonometric specializations *J MATH PHYS*, **40**, 1627-1663, 1999

**Ruijsenaars SNM** Integrable particle systems vs solutions to the KP and 2D Toda equations *ANN PHYS-NEW YORK*, **256**, 226-301, 1997

**Ruijsenaars SNM** First order analytic difference equations and integrable quantum systems *J MATH PHYS*, **38**, 1069-1146, 1997

**RUIJSENAARS S** ACTION-ANGLE MAPS AND SCATTERING-THEORY FOR SOME FINITE-DIMENSIONAL INTEGRABLE SYSTEMS .3. SUTHERLAND TYPE SYSTEMS AND THEIR DUALS *PUBL RES I MATH SCI*, **31**, 247-353, 1995

**RUIJSENAARS SNM** ACTION-ANGLE MAPS AND SCATTERING-THEORY FOR SOME FINITE-DIMENSIONAL INTEGRABLE SYSTEMS .2. SOLITONS, ANTISOLITONS, AND THEIR BOUND-STATES *PUBL RES I MATH SCI*, **30**, 865-1008, 1994

**RUIJSENAARS SNM** RELATIVISTIC TODA SYSTEMS *COMMUN MATH PHYS*, **133**, 217-247, 1990

**RUIJSENAARS SNM** INDEX FORMULAS FOR GENERALIZED WIENER-HOPF OPERATORS AND BOSON-FERMION CORRESPONDENCE IN 2N DIMENSIONS *COMMUN MATH PHYS*, **124**, 553-593, 1989

**RUIJSENAARS SNM** ACTION-ANGLE MAPS AND SCATTERING-THEORY FOR SOME FINITE-DIMENSIONAL INTEGRABLE SYSTEMS .1. THE PURE SOLITON CASE *COMMUN MATH PHYS*, **115**, 127-165, 1988

**CAREY AL, RUIJSENAARS SNM** ON FERMION GAUGE GROUPS, CURRENT-ALGEBRAS AND KAC-MOODY ALGEBRAS *ACTA APPL MATH*, **10**, 1-86, 1987

**FULLING SA, RUIJSENAARS SNM** TEMPERATURE, PERIODICITY AND HORIZONS *PHYS REP*, **152**, 135-176, 1987

**RUIJSENAARS SNM** COMPLETE-INTEGRABILITY OF RELATIVISTIC CALOGERO-MOSER SYSTEMS AND ELLIPTIC FUNCTION IDENTITIES *COMMUN MATH PHYS*, **110**, 191-213, 1987

**RUIJSENAARS SNM, SCHNEIDER H** A NEW CLASS OF INTEGRABLE SYSTEMS AND ITS RELATION TO SOLITONS *ANN PHYS-NEW YORK*, **170**, 370-405, 1986

**CAREY AL, RUIJSENAARS SNM, WRIGHT JD** THE MASSLESS THIRRING MODEL - POSITIVITY OF KLAIBER N-POINT FUNCTIONS *COMMUN MATH PHYS*, **99**, 347-364, 1985

**RUIJSENAARS SNM** THE AHARONOV-BOHM EFFECT AND SCATTERING-THEORY *ANN PHYS-NEW YORK*, **146**, 1-34, 1983

**RUIJSENAARS SNM** ON THE 2-POINT FUNCTIONS OF SOME INTEGRABLE RELATIVISTIC QUANTUM-FIELD THEORIES *J MATH PHYS*, **24**, 922-931, 1983

**Ruijsenaars SNM** On the two-point functions of some integrable relativistic quantum field theories *Journal of Mathematical Physics*, **24**, 922-931, 1982

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**RUIJSENAARS SNM** THE WIGHTMAN AXIOMS FOR THE FERMIONIC FEDERBUSH MODEL *COMMUN MATH PHYS*, **87**, 181-228, 1982

**RUIJSENAARS SNM** SCATTERING-THEORY FOR THE FEDERBUSH, MASSLESS THIRRING AND CONTINUUM ISING-MODELS *J FUNCT ANAL*, **48**, 135-171, 1982

**RUIJSENAARS SNM** ON NEWTON-WIGNER LOCALIZATION AND SUPERLUMINAL PROPAGATION SPEEDS *ANN PHYS-NEW YORK*, **137**, 33-43, 1981

**RUIJSENAARS SNM** INTEGRABLE QUANTUM FIELD-THEORIES AND BOGOLIUBOV TRANSFORMATIONS *ANN PHYS-NEW YORK*, **132**, 328-382, 1981

**RUIJSENAARS SNM** ON ONE-DIMENSIONAL INTEGRABLE QUANTUM-SYSTEMS WITH INFINITELY MANY DEGREES OF FREEDOM *ANN PHYS-NEW YORK*, **128**, 335-362, 1980

**RUIJSENAARS SNM** THE CONTINUUM LIMIT OF THE INFINITE ISOTROPIC HEISENBERG CHAIN IN ITS GROUND-STATE REPRESENTATION *J FUNCT ANAL*, **39**, 75-84, 1980

**HEGERFELDT GC, RUIJSENAARS SNM** REMARKS ON CAUSALITY, LOCALIZATION, AND SPREADING OF WAVE-PACKETS *PHYS REV D*, **22**, 377-384, 1980

**RUIJSENAARS SNM** A POSITIVE ENERGY DYNAMICS AND SCATTERING-THEORY FOR DIRECTLY INTERACTING RELATIVISTIC-PARTICLES *ANN PHYS-NEW YORK*, **126**, 399-449, 1980

**RUIJSENAARS SNM** GAUGE-INVARIANCE AND IMPLEMENTABILITY OF THE S-OPERATOR FOR SPIN-O AND SPIN-1/2 PARTICLES IN TIME-DEPENDENT EXTERNAL FIELDS *J FUNCT ANAL*, **33**, 47-57, 1979

**GARBER WD, RUIJSENAARS SNM, SEILER E, BURNS D** FINITE ACTION SOLUTIONS OF THE NON-LINEAR SIGMA-MODEL *ANN PHYS-NEW YORK*, **119**, 305-325, 1979

**Garber WD, Ruijsenaars SNM, Seiler E, Burns D** On finite action solutions of the nonlinearσ-model *Annals of Physics*, **119**, 305-325, 1979

DOI:10.1016/0003-4916(79)90189-1

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**Ruijsenaars SNM** On Bogoliubov transformations. II. The general case *Annals of Physics*, **116**, 105-134, 1978

DOI:10.1016/0003-4916(78)90006-4

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**RUIJSENAARS SNM** BOGOLIUBOV TRANSFORMATIONS .2. GENERAL CASE *ANN PHYS-NEW YORK*, **116**, 105-134, 1978

**RUIJSENAARS SNM, BONGAARTS PJM** SCATTERING THEORY FOR ONE-DIMENSIONAL STEP POTENTIALS *ANN I H POINCARE A*, **26**, 1-17, 1977

**RUIJSENAARS SNM** CHARGED-PARTICLES IN EXTERNAL FIELDS .1. CLASSICAL-THEORY *J MATH PHYS*, **18**, 720-737, 1977

**RUIJSENAARS SNM** CHARGED-PARTICLES IN EXTERNAL FIELDS .2. QUANTIZED DIRAC AND KLEIN-GORDON THEORIES *COMMUN MATH PHYS*, **52**, 267-294, 1977

**RUIJSENAARS SNM** BOGOLIUBOV TRANSFORMATIONS FOR SYSTEMS OF RELATIVISTIC CHARGED-PARTICLES *J MATH PHYS*, **18**, 517-526, 1977

**Ruijsenaars SNM** On Bogoliubov transformations for systems of relativistic charged particles *Journal of Mathematical Physics*, **18**, 517-526, 1976

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**BONGAARTS PJM, RUIJSENAARS SNM** KLEIN PARADOX AS A MANY-PARTICLE PROBLEM *ANN PHYS-NEW YORK*, **101**, 289-318, 1976

**BERENDS FA, RUIJSENA.SN** EXAMPLES OF PHASE-SHIFT AMBIGUITIES FOR SPINLESS ELASTIC-SCATTERING *NUCL PHYS B*, **B 56**, 507-524, 1973

**BERENDS FA, RUIJSENA.SN** DIFFERENT SETS OF PHASE-SHIFTS FOR SAME DIFFERENTIAL CROSS-SECTION AND POLARIZATION IN SPIN-O-SPIN-1/2 ELASTIC-SCATTERING *NUCL PHYS B*, **B 56**, 525-535, 1973

**Hallnäs M, Ruijsenaars S** Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems *J Math Phys 53, 123512 (2012) 64 pages*

**Ruijsenaars S, Hallnas M** Product formulas for the relativistic and nonrelativistic conical functions *Advanced Studies in Pure Mathematics*

View abstract

**Hallnas M, Ruijsenaars S** Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. II. The two- and three-variable cases *International Mathematics Research Notices*

DOI:10.1093/imrn/rnx020

View abstract

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