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Book of Abstracts 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 < 1, 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
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