Jairo Charris Seminar 2010 - Scientific Program

Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics

Schedule

Time	Wednesday – Chairman:	Thursday – Chairman:	Friday	Saturday – Chairman:
8h30	Registraton	Plenary Talk: V. Sokolov	Social meeting-Visit to Tayrona Natural Park	Plenary Talk: V. Ovsienko
9h30	Welcoming	Plenary Talk: V. Sokolov		Plenary Talk: V. Ovsienko
10h00	Break	Break		Break
10h30	Plenary Talk: V. Spiridonov	Plenary Talk: R. Milson		Poster Sesion
10h30	Plenary Talk: V. Spiridonov	Plenary Talk: R. Milson		Open questions and future directions
12h00	Lunch	Lunch		Special Lunch
14h	Talk: L. Garza	Talk: V. Hussin
	Break	Break
15h	Talk: J. M. Guilarte	Talk: J.J. Morales-Ruiz

Conferences

Title: Spectral transformations of orthogonal polynomials: A connection to integrable systems.

Abstract: Orthogonal polynomials with respect to measures supported on the real line satisfy a three term recurrence relation that can be represented on matrix form by means of a symmetric tri-diagonal matrix known as Jacobi matrix. When certain spectral transformations are applied to the orthogonality measure, it is possible to obtain the Jacobi matrix associated to the perturbed measure using LU and UL factorizations of the original Jacobi matrix. These factorizations are also known in the literature as Darboux transformations. Analogously, spectral transformations to measures supported on the unit circle can be expressed by means of (almost) QR factorizations of a Hessenberg matrix. In this contribution, we present some results in this direction and their relation with some integrable systems when a temporal variable is introduced on the entries of such matrices. Some interesting open problems will be brieﬂy discussed.

Key words: Spectral transformations, Orthogonal polynomials, Jacobi matrices, Darboux transformations, Integrable systems.

Title: Two-dimensional Morse potentials and quantum coherent and squeezed state

Abstract: For one dimensional non relativistic quantum systems, coherent and squeezed states have been largely studied, starting from those of the harmonic oscillator. They have been used in order to understand the quantum properties of light. They have also been generalized in different ways using methods of group and supergroup theories. In particular, we have recently been able to construct different types of those states for the completely solvable system of a one-dimensional anharmonic oscillator. Such a system represents well the anharmonic vibrations in diatomic molecules and the potential associated is the well-known Morse potential. Properties of those states in terms of squeezing, minimum Heisenberg uncertainty principle, space localization and statistics have been studied.

Solvable higher dimensional quantum systems have been classified using different approaches like algebraic, factorization, shape invariance and also supersymmetric methods. In particular, the spectra of partner Hamiltonians for the Morse potential in two dimensions have been studied using supersymmetry and second order supercharges. We thus have extended the construction of coherent and squeezed states to these systems. Since, such systems present degeneracies in their spectra, those states must be carefully defined. As in the one dimensional case, good spatial localization of those states are obtained by adjusting the coherent and squeezing parameters.

Title: Supersymmetric Quantum Mechanics based on Classical Integrable Systems

Abstract: Starting from the Kepler (one Newtonian center, in various dimensions and topologies) and the Euler (two planar Newtonian centers) problems we will describe how Supersymmetric Quantum Mechanical systems can be constructed. In the SUSY Kepler problem the spectrum can be obtained by algebraic means with the help of the Runge-Lenz integral of motion. In the other case, the SUSY two center problem, the spectrum is determined from the QES Razavy and Whittaker-Hill equations.

Abstract: We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described exceptional polynomial families in terms of an isospectral Darboux fransformation. The latter families are characterized as complete orthogonal polynomial systems $p_m, p_{m+1}, \ldots $ that are defined by a second order eigenvalue equations, and where $\deg p_j = j$ and $m>0$. Like the classical families, these orthogonal polynomials admit raising and lowering operators. We show that this shape-invariance property is a direct consequence of the permutability property of the Darboux-Crum transformation.

Abstract: In this work we will show that the Galois Theory of Linear Differential Equations, also called the Picard-Vessiot Theory, is the natural framework to study the solvability in closed form of the Schrodinger equation of Quantum Mechanics. Thus, all the cases where this equation was solved by quadratures can be classiffied using the Picard-Vessiot Theory. As a by product, we obtain a Galoisian interpretation of the Supersymmetric Quantum Mechanics. This is a joint work with Jacques-Arthur Weil and Primitivo B. Acosta-Humanez. A preliminary version of this work can be found at http://arxiv.org/abs/0906.3532.

Abstract: The pentagram map is a projectively natural iteration defined on polygons. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is discrete completely integrable system. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.

Abstract: Given an associative multiplication in the matrix algebra compatible with the usual one or, in other words, a linear associative deformation of the matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call $M$-structures. We describe an important class of $M$-structures. The classification of these $M$-structures naturally leads to affine Dynkin diagrams of $A$,

$D$, $E$-type. These $M$-structures and their representations can be described in terms of quiver representations. We investigate in details the multiplications of the $\tilde A_{2 k-1}$-type and integrable matrix ODEs and PDEs generated by them.

Speaker: V.P. Spiridonov (Laboratory of Theor. Physics, JINR, Dubna, Russia)

Title: Elliptic hypergeometric functions in integrable systems and superconformal field theories.

Abstract: Elliptic hypergeometric functions form a new class of special functions of mathematical physics. We shall briefly survey their appearance in integrable systems in the context of elliptic solutions of the Yang-Baxter equation, solutions of L-A pair equations for a generalized discrete time chain, and finite difference Calogero-Sutherland type models. However, the main attention will be paid to recent supersymmetric interpretation of the elliptic hypergeometric integrals as superconformal (topological) indices of supersymmetric field theories. Connections with the supersymmetric quantum mechanics, Witten index, and the Darboux type transformations will be mentioned in appropriate places.