Dynamical systems on infinitely sheeted Riemann surfaces

This paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi-valuedness of the solutions as functions of complex time tau. In this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surface R -> C = {tau} is known to be an infinitely sheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the tau-plane is dense. The main novelty of this paper is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows us to study global properties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions. The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real time trajectories of the system are given by paths on R that are projected to a circle on the complex plane tau. Due to the branching of R, the solutions may have different periods or may be aperiodic

Quasi-exact solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: we show that the generalized Lame potential possesses four algebraic sectors, and describe a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic

Oscillation Theorems for the Wronskian of an Arbitrary Sequence of Eigenfunctions of Schrodinger's Equation

The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schrodinger's equation to have constant sign in its domain of definition. We extend this result by giving explicit formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions. Our results apply in particular to Wronskians of classical orthogonal polynomials, thus generalizing classical results by Karlin and SzegA. Our formulas hold under very mild conditions that are believed to hold for generic values of the parameters. In the Hermite case, our results allow to prove some conjectures recently formulated by Felder et al

Recurrence relations for exceptional Hermite polynomials.

The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in x

A_(N)-type Dunkl operators and new spin Calogero-Sutherland models

A new family of AN-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero-Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians

Asymptotically isochronous systems

Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotic ally to functions periodic with the sa m e fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrable indeed solvable many-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrable and isochronous) many-body problem with inverse-cube two-body forces and a one-body linear ("harmonic oscillator") force

An extension of Bochner's problem: exceptional invariant subspaces

We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of polynomial eigenfunctions p(1), p(2),.. of all degrees except for degree zero. The main theorem of the paper provides a characterization of all such differential operators. The existence of such differential operators and polynomial sequences is based on the concept of exceptional polynomial subspaces, and the converse part of the main theorem rests on the classification of codimension one exceptional subspaces under projective transformations, which is performed in this paper

Two-step Darboux transformations and exceptional Laguerre polynomials

It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre and the type II X(l)-Laguerre polynomials are recovered

Scaling, stability and distribution of the high-frequency returns of the IBEX35 index

© 2012 Elsevier B.V. The research of DGU has been supported in part by Direccion General de Investigacion, Ministerio de Ciencia e Innovacion of Spain, under grants MTM2009-06973 and MTM2012-31714In this paper we perform a statistical analysis of the high-frequency returns of the IBEX35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data using maximum likelihood estimation and different goodness-of-fit tests rejects the Levy-stable law as a plausible underlying probabilistic model. The analysis shows that the Normal Inverse Gaussian distribution provides an overall fit for the data better than any of the other subclasses of the family of Generalized Hyperbolic distributions and certainly much better than the Levy-stable laws. Furthermore, the right (resp. left) tail of the distribution seems to follow a power-law with exponent alpha approximate to 4.60 (resp. alpha approximate to 4.28). Finally, we present evidence that the observed stability is due to temporal correlations or non-stationarities of the data.Direccion General de Investigacion, Ministerio de Ciencia e Innovacion of SpainDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Complete classification of rational solutions of A(2n)-Painleve systems.

We provide a complete classification and an explicit representation of rational solutions to the fourth Painleve equation P-IV and its higher order generalizations known as the A(2n)-Painleve or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrodinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the A(2n)-Painleve system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Backlund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles. (C) 2021 Elsevier Inc. All rights reserved