Book of Abstracts International Workshop onSymmetries, Special Functions andSuperintegrability

A conference in honor of Prof. Mariano A. del Olmo on the occasion of his 60th birthday

July 10–11, 2014 Scientific Committee

Angel´ Ballesteros (Burgos) Jose´ F. Carinena˜ (Zaragoza) Enrico Celeghini (Firenze) Pilar G. Estevez´ (Salamanca) David J. Fernandez´ (Mexico´ DF) Javier Negro (Valladolid) Miguel A. Rodr´ıguez (Madrid) Mariano Santander (Valladolid)

Organizing Committee

Oscar´ Arratia Juan A. Calzada Fernando Gomez-Cubillo´ Manuel Gadella Jose´ M. Izquierdo S¸engul¨ Kuru Javier Negro Luis M. Nieto Mariano Santander Foreword

The “International Workshop on Symmetries, Special functions and Superintegrability” (S-3) is organized to celebrate the 60th birthday of Professor Mariano del Olmo, an internation- ally recognized expert in many branches of mathematical physics. He has made remarkable contributions in this field since the the beginning of the eighties: quantum groups, coherent states, integrable and superintegrable systems, representations of Lie groups and Lie alge- bras, special functions and other topics that show his wide range of interest. The workshop program is precisely centered around some of these topics in which Ma- riano has been working for many years. It is a pleasure to see that many of his present and former collaborators, either inside Spain or working in other countries, are making an effort to come to Valladolid and share with him this unique occasion.

The Organizing Committee

Valladolid, July 10th, 2014.

1 Contents

Symmetries from the Solution Manifold Victor Aldaya ...... 3 Time asymmetric quantum mechanics Arno Bohm ...... 3 A generalized approach to integrability by quadratures Jose´ F. Carinena˜ ...... 4 Penning trap in a rotating magnetic field: coherent states approach David J. Fernandez´ ...... 4 1 + 1 spectral problems arising from the Manakov-Santini system Pilar G. Estevez...... ´ 4 Exceptional orthogonal polynomials David Gomez–Ullate´ ...... 5 On the planar Demkov wave functions Miguel A. Gonzalez´ Leon...... ´ 5 Natural Limits of Electroweak Model and Contraction of its Gauge Group Nikolai Gromov ...... 5 Kac-Moody-Virasoro symmetries of variable coefficient nonlinear evolution equations in 2 + 1 dimensions Faruk Gung¨ or...... ¨ 6 N−1 Invariant solutions of the supersymmetric CP sigma model Veronique´ Hussin ...... 6 On the structure of Schwinger’s measurement algebra: groups, groupoids and 2-groupoids Alberto Ibort ...... 6 Vacuum Carlos Lopez-Lacasta´ ...... 7 Scalar field fluctuations distorted by two pairs of δ − δ0 interactions Juan Mateos Guilarte ...... 7 Supersymmetric partners of the truncated harmonic oscillator Vicente S. Morales ...... 7 Exotic supersymmetry of reflectionless systems, and solitons Mikhail Plyushchay ...... 8 Some new superintegrable hamiltonian systems, solvable by factorization method Orlando Ragnisco ...... 8 Invariant transformations in Euclidean hyperkahler structures Miguel A. Rodr´ıguez ...... 8 The role of indencomposable representations in statistical physics Yvan Saint–Aubin ...... 8 Haantjes Manifolds and Integrable Systems Piergiulio Tempesta ...... 9 The WKB approximation in deformation quantization Jaromir Tosiek ...... 9 Position and momentum in the monochromatic Maxwell fish-eye Kurt Bernardo Wolf ...... 9

2 List of invited talks

Symmetries from the Solution Manifold Victor Aldaya

Instituto de Astrof´ısica de Andaluc´ıa (Spain)

In this talk, extended symmetries (non-point symmetries) of a physical system are used to characterize the corresponding solution manifold by means of Noether invariants. This is the starting point to the correct quantization in non-linear cases, where the success of Canonical Quantization is not guaranteed. The use of the Poincare-Cartan´ form permits finding both the symplectic structure on the solution manifold, through the Hamilton-Jacobi transformation, and the required symmetries, realized as Hamiltonian vector fields, associated with functions on the solution manifold (thus constituting an inverse of the Noether Theorem) lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping. We present simple non-trivial examples where the symmetries are found in a perturbative way.

Time asymmetric quantum mechanics Arno Bohm

The University of Texas at Austin (USA)

The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrodinger¨ equation for states or the Heisenberg equation for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time sym- metric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration appa- ratus (detector)). If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = ~/Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS’s of Hardy functions and connected with it, to a semigroup time evolution t0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum me- chanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.

3 A generalized approach to integrability by quadratures Jose´ F. Carinena˜

Universidad de Zaragoza (Spain)

The classical result of Lie on integrability by quadratures will be reviewed and some generalizations will be proposed. After a short review of the classical Lie theorem it will be shown that if we are able to construct in an iterative way a nested sequence of subalgebras Li of vector fields including the dynamical vector field and after some steps the Lie subalgebra is Abelian, then the dynamics is integrable by k−1 quadratures. The theory will be illustrated with examples and an extension of the theorem where the Lie algebras are replaced by some distributions will also be presented.

Penning trap in a rotating magnetic field: coherent states approach David J. Fernandez´

CINVESTAV (Mexico)´

The quantum problem of a non-relativistic charged particle subject to the electric and magnetic fields created by an ideal Penning trap plus a magnetic field rotating around the symmetry axis is addressed. The transition to the rotating frame is performed in order to eliminate the initial time-dependence in the Hamiltonian; then a coherent states approach for the resulting time-independent problem is implemented. It is determined the parameters domain for the particle to be in the trap regime. Several physical quantities for the system being in a coherent state are also evaluated.

1 + 1 spectral problems arising from the Manakov-Santini system Pilar G. Estevez´

Universidad de Salamanca (Spain)

This talk deals with the spectral problem of the Manakov Santini system. The point Lie symmetries of the Lax pair have been identified. Several similarity reductions arise from these symmetries. An important benefit of our procedure is that the study of the Lax pair instead of the partial differential equations yields the reductions of the eigenfunctions and also the spectral parameter. Therefore, we have obtained five interesting spectral problems in 1 + 1 dimensions.

4 Exceptional orthogonal polynomials David Gomez–Ullate´

Universidad Complutense (Spain)

Exceptional orthogonal polynomials are extensions of classical orthogonal polynomials in the sense that they are also eigenfunctions of a Sturm-Liouville problem although the degree sequence has a finite number of gaps (missing degrees). Equivalently, in mathematical physics they appear as eigenfunctions of rational extensions of exactly solvable potentials. It has been conjectured that all exceptional orthogonal polynomials can be obtained through Darboux-Crum transformations of their classical counterparts, and this conjecture has been proved in the Hermite case using the connection between trivial monodromy potentials and Darboux transformations. In this talk we will review some of these recent results and we will mention current open problems.

On the planar Demkov wave functions Miguel A. Gonzalez´ Leon´

Universidad de Salamanca (Spain)

The quantum spectral problem of diatomic molecular ions, in the Born-Oppenheimer approximation, is separable using spheroidal coordinates obtained by rotating the two-dimen- sional elliptic coordinates about the focal axis. Separability allows to split the Schrodinger¨ equation in a system of ODEs consisting in two different Generalized Spheroidal equations for the three dimensional case, and the Razavy equation plus the Whittaker Hill equation in the two dimensional one. Demkov has searched for eigen-wave functions of the Hamiltonian corresponding to the energy levels of a hydrogenoid atom. This procedure lead, for certain values of the nuclear charges, to the construction of “finite” wave functions, the so-called Demkov wave functions. We have already explained this behaviour in terms of the Quasi-Exact Solvability of the underlying equation, the Confluent Heun equation. We present here a complete analysis of the analogous planar Demkov wave functions.

Natural Limits of Electroweak Model and Contraction of its Gauge Group Nikolai Gromov

Syktyvkar University (Russian Federation)

Energy is the natural and most importantt parameter in particle physics. The Electroweak Model is a gauge theory based on the group SU(2) × U(1), acting in the boson, lepton and quark sectors, which are describes by vectors of C2 with different physical interpretations in

5 different sectors. The contracted group SU(2; j) and its fundamental representation space C2(j) are obtained by the consistent rescaling of SU(2) and C2

0 ! ! ! jz α jβ jz1 z0(j) = 1 = = u(j)z(j) 0 ¯ z2 −jβ α¯ z2 when j → o. Transformation rules of the boson felds as well as of the left lepton and quark ± ± fields are as follows: Wµ → jWµ , Zµ → Zµ, Aµ → Aµ, νl → jνl, el → el, ul → jul, dl → dl. The right lepton and quark fields are SU(2)-singlets, i.e. scalars, and therefore are not transformed. The parameter j is connected with the energy s in center-of-mass system 2 √ j (s) = g s/mW , where mW is W -boson mass and g is constant. So contraction j → 0 corresponds to zero energy limit of the Electroweak Model.

Kac-Moody-Virasoro symmetries of variable coefficient nonlinear evolution equations in 2 + 1 dimensions Faruk Gung¨ or¨

Istambul Technical University (Turkey)

In this talk I will consider variable coefficient generalizations of two well-known physi- cal models: Kadomtsev-Petviashvili (KP) and Zabolotskaya-Khoklov (or dispersionless KP) equation and discuss how the existence of Kac-Moody-Virasoro algebras as their point sym- metry algebras, when combined with equivalence transformations, can serve as an additional test for integrability.

N−1 Invariant solutions of the supersymmetric CP sigma model Veronique´ Hussin

Universite´ de Montreal´ (Canada)

Constant curvature surfaces are constructed from the finite action solutions of the super- N−1 symmetric CP sigma model. They are shown to be related to invariant solutions of the model. In the case of holomorphic solutions, we get an unicity theorem. Some new results are obtained for the case of non-holomorphic solutions.

On the structure of Schwinger’s measurement algebra: groups, groupoids and 2-groupoids Alberto Ibort

Universidad Carlos III de Madrid (Spain)

The structure of Schwinger’s measurement algebra will be reviewed. We would point out that it carries the structure of a 2-groupoid and the basic notions of groups, gropoids and 2-groupoids will be discussed as well as some elementary examples.

6 Vacuum Carlos Lopez-Lacasta´

Universidad de Alcala´ de Henares (Spain)

Vacuum as a Lorentz invariant fluid (distribution of density of particles in momentum space), equivalently a relativistic ether, is the subject of this talk. In QM, the vacuum of vir- tual particles will be considered in a theory of hidden variables, not as a scientific proposal (because of the uncertainty principle), but as a (metaphysical) analogy between some QM phenomena and classical Brownian motion. In GR, there is wide agreement that the Cosmo- logical constant, a property of spacetime vacuum, is the source of dark energy. Although the existence of exotic particles is the main research line in the dark matter issue, quintessence theories, grounded in some new cosmological field, have the advantage of unifying dark mat- ter and dark energy in a single phenomenon. Vacuum inhomogeneity at a cosmological scale is suggested as a candidate for quintessence; a brief qualitative analysis of a specific model will be presented.

Scalar field fluctuations distorted by two pairs of δ − δ0 interactions Juan Mateos Guilarte

Universidad de Salamanca (Spain)

Scattering solutions by a double δ − δ0 will be used as one-particle solutions of an scalar field quantum field theory in a (1 + 1)-dimensional Minkowskian space-time. Following the trend of mimicking the conducting plates used in a Casimir experiment set-up by means of point δ interactions I will discuss the quantum vacuum interaction between plates relying on the Lipmann-Schwinger T -matrix.

Supersymmetric partners of the truncated harmonic oscillator Vicente S. Morales

CINVESTAV-IPN (Mexico)´

Supersymmetry transformations of first and second order are used to generate Hamilto- nians with known spectra departing from the harmonic oscillator with an infinite potential barrier. Certain systems obtained in a straightforward way through said procedure possess differential ladder operators of both types, third and fourth order. Since systems with this kind of operators are linked with the Painleve IV and Painleve V equations respectively, sev- eral solutions of these non-linear second-order differential equations will be simply found, along with a chain of Backlund transformations connecting such solutions.

7 Exotic supersymmetry of reflectionless systems, and solitons Mikhail Plyushchay

Universidad de Santiago de Chile (Chile)

Finite-gap and reflectionless quantum systems are characterized by exotic supersymme- try. We review the peculiarities of such supersymmetric structure appearing in reflectionless systems associated with the soliton and multi-kink-antikink solutions of the KdV and mKdV equations. We also discuss the transmutations of supersymmetry related to the soliton scat- tering.

Some new superintegrable hamiltonian systems, solvable by factorization method Orlando Ragnisco

Universita` di Roma 3 (Italy)

Following a recent paper by Kuru and Negro, we apply the “spectrum generating algebra” method (in the classical case) and the factorisation method (in the quantum case) to solve a pair of new hamiltonian systems, Taub-Nut and Darboux III, which are superintegrable deformations on curved spaces of the Kepler-Coulomb and Harmonic Oscillator.

Invariant transformations in Euclidean hyperkahler structures Miguel A. Rodr´ıguez

Univesidad Complutense de Madrid (Spain)

We present a detailed construction of the invariance algebra of the quaternionic structure in Euclidean spaces. Related results, as maximal subgroups of simple Lie groups and the Berger’s list of holonomy groups, will be also discussed.

The role of indencomposable representations in statistical physics Yvan Saint–Aubin

Universite´ de Montreal´ (Canada)

The transfer matrix is a common tool in statistical physics and shares many properties with the Hamiltonian in quantum physics. However, contrarily to Hamiltonians, transfer ma- trices do not need to be Hermitian. They do not even need to have real eigenvalues. This distinction is of physical importance and efforts have been given to understand the representa- tion theory of algebras arising in lattice models like, for example, the families of Temperley- Lieb algebras (TL algebras). One telltale signature of this distinction is the existence of

8 Jordan blocks in transfer matrices and, more generally, of indecomposable representations. I shall give a physical example of these Jordan blocks and report on the efforts to classify all indecomposable representations of the TL algebras and dilute TL algebras.

Haantjes Manifolds and Integrable Systems Piergiulio Tempesta

Universidad Complutense (Spain)

A general theory of integrable systems is proposed, based on the theory of Haantjes manifolds. We introduce the notion of symplectic-Haantjes manifold (or ωH manifold), as the natural setting where the notion of integrability can be formulated. We propose an ap- proach to the separation of variables for classical systems, related to the geometry of Haantjes manifolds. A special class of coordinates, called Darboux-Haantjes coordinates, will be con- structed from the Haantjes structure associated with an integrable systems. They enable the additive separation of variables of the Hamilton-Jacobi equation. We also present an ap- plication of our approach to the study of some finite-dimensional integrable models, as the Henon-Heiles´ systems and a stationary reduction of the KdV hierarchy.

The WKB approximation in deformation quantization Jaromir Tosiek

Technical University of Lodz (Poland)

An adaptation of the WKB approximation to the deformation quantization is presented. A relationship between the phase σ(~r) of a wave function exp i σ(~r)
and a respective ~ Wigner function is derived. Formulas for a Wigner function of a wave function being a product of functions and a superposition of functions are proposed. An example of the semi- classical approximation in deformation quantization is analysed.

Position and momentum in the monochromatic Maxwell fish-eye Kurt Bernardo Wolf

Universidad Nacional Autonoma´ de Mexico´ (Mexico)´

In geometric optics the Maxwell fish-eye is a medium where light rays follow circles, while in scalar wave optics this medium can only ‘trap’ fields of certain discrete frequencies. In the monochromatic case characterized by a positive integer `, there are 2`+1 independent fields. We identify two bases of functions that serve as wavefields of definite position and definite momentum. Their construction uses the stereographic projection of the sphere, and the identification is corroborated in the ` → ∞ limit to a homogeneous Helmholtz medium.

9 Morning Session, 10 July

10:00- 10:30 Opening Session

Chairman: J.F. Carinena˜

10:30 - 11:00 V. Aldaya Symmetries from the solution manifold 11:00 - 11:30 M.A. Rodr´ıguez Invariant transformations in Euclidean hy- perkahler structures 11:30- 12:15 Coffee Break

Chairman: D.J. Fernandez´

12:15 - 12:45 A. Bohm Time asymmetric quantum mechanics 12:45 - 13:15 O. Ragnisco Some new superintegrable hamiltonian sys- tems, solvable by factorization method 13:15 - 13:45 Y. Saint-Aubin The role of indencomposable representations in statistical physics 13:45 - 14:15 M. Plyushchay Exotic supersymmetry of reflectionless sys- tems, and solitons 14:15- 16:00 Lunch

Afternoon Session, 10 July

Chairman: M.A. Rodr´ıguez

16:00 - 16:30 P.G. Estevez´ 1+1 spectral problems arising from the Manakov-Santini system 16:30 - 17:00 P. Tempesta Haantjes manifolds and integrable systems 17:00 - 17:30 F. Gung¨ or¨ Kac-Moody-Virasoro symmetries of variable coefficient nonlinear evolution equations in 2 + 1 dimensions 17:30- 18:00 Coffee Break

Chairman: A. Ballesteros

18:00 - 18:30 J. Tosiek The WKB approximation in deformation quantization 18:30 - 19:00 M.A. Gonzalez-Le´ on´ On the planar Demkov wave functions 19:00 - 19:30 D. Gomez-Ullate´ Exceptional orthogonal polynomials 19:30 - 20:00 V.S. Morales-Salgado Supersymmetric partners of the truncated harmonic oscillator

10 Morning Session, 11 July

Chairman: M. Santander

09:00 - 9:30 K.B. Wolf Position and momentum in the monochro- matic Maxwell fish-eye 9:30 - 10:00 D.J. Fernandez´ Penning trap in a rotating magnetic field: co- herent states approach 10:00 - 10:30 V. Hussin Invariant solutions of the supersymmetric N−1 CP sigma model 10:30 - 11:00 N. Gromov Natural limits of electroweak model and con- traction of its gauge group 11:00- 11:45 Coffee Break

Chairman: E. Celeghini

11:45 - 12:15 J. Mateos-Guilarte Scalar field fluctuations distorted by two pairs of δ − δ0 interactions 12:15 - 12:45 C. Lopez´ Vacuum 12:45 - 13:15 A. Ibort On the structure of Schwinger’s measurement algebra: groups, groupoids and 2-groupoids 13:15 - 13:45 J.F. Carinena˜ A generalized approach to integrability by quadratures 13:45- 14:00 Closing Session

Wifi Access: SSID: workshopmolmo Password: mdelolmo

11 Participants

Page

Aldaya,Victor [email protected] 3 Alvarez,´ Juan J. [email protected] Arratia, Oscar´ [email protected] Aslar, Engin [email protected] Ballesteros, Angel´ [email protected] Bohm, Arno [email protected] 3 Calzada, Juan A. [email protected] Campoamor-Stursberg, Rutwig [email protected] Carinena,˜ Jose´ F. [email protected] 4 Celeghini, Enrico celeghini@fi.infn.it Cervero,´ Jose´ M. [email protected] C¸evik, Dogukanˇ [email protected] de la Torre, Marina [email protected] D´ıaz-Bautista, Erik ediaz@fis.cinvestav.mx Fernandez,´ David J. david@fis.cinvestav.mx 4 Finkel, Federico ffinkel@fis.ucm.es Gadella, Manuel [email protected] Estevez,´ Pilar G. [email protected] 4 Gomez-Ullate,´ David [email protected] 5 Gomez-Cubillo,´ Fernando [email protected] Gonzalez´ Leon,´ Miguel A. [email protected] 5 Gonzalez-L´ opez,´ Artemio artemio@fis.ucm.es Gromov, Nikolai [email protected] 5 Guerrero, Julio [email protected] Gung¨ or,¨ Faruk [email protected] 6 Herranz, Francisco J. [email protected] Hussin, Veronique´ [email protected] 6 Ibort, Alberto [email protected] 6 Izquierdo, Jose´ M. [email protected] Kuru, S¸engul¨ [email protected] Lopez-Lacasta,´ Carlos [email protected] 7 Mateos Guilarte, Juan [email protected] 7 Morales, Vicente S. vmorales@fis.cinvestav.mx 7 Munoz,˜ Miguel C. [email protected] Negro, Javier [email protected] Nieto, Luis M. [email protected] Olmo, Mariano A. del [email protected] Pascual, Jose´ F. [email protected] Plyushchay, Mikhail [email protected] 8 Ragnisco, Orlando ragnisco@fis.uniroma3.it 8 Rodr´ıguez, Miguel A. rodrigue@fis.ucm.es 8 Saint-Aubin, Yvan [email protected] 8 Santander, Mariano [email protected] Tempesta, Piergiulio p.tempesta@fis.ucm.es 9 Tosiek, Jaromir [email protected] 9 Wrickramasekara, Sujeev [email protected] Wolf, Kurt B. bwolf@fis.unam.mx 9

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