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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

Home page

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
View abstract

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
View abstract

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
View abstract

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
View abstract

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 CaseSYMMETRY 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 polynomialsRAMANUJAN JOURNAL, 339-364 2007
DOI:10.1007/s11139-006-0257-x
View abstract

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
View abstract

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 functionJOURNAL 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 operatorsJOURNAL 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 EigenfunctionsNEW 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 revisitedJOURNAL 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 reflectionTHEORETICAL 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 conjecturesJOURNAL 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 equationsSPECIAL FUNCTIONS 2000: CURRENT PERSPECTIVE AND FUTURE DIRECTIONS, 281-333 2001

Ruijsenaars SNM Sine-Gordon solitons vs. relativistic Calogero-Moser particlesINTEGRABLE 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