arXiv subjects

David Gomez-Ullate

Publications and source records attributed to David Gomez-Ullate.

22 records · Page 2Linked to original sources

arXiv2006

Quasi-exact solvability beyond the SL(2) algebraization

We present evidence to suggest that the study of one dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual $\sla(2)$ approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the $\sla(2)$ Lie algebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic hamiltonian cannot be expressed as a polynomial in the generators of $\sla(2)$. We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.

nlin.SI↗

arXiv2006

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 $τ$. 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 ${\mathcal R}$ is known to be an infinitely sheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the $τ$-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 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 $\mathcal R$ that are projected to a circle on the complex plane $τ$. Due to the branching of $\mathcal R$, the solutions may have different periods or may not be periodic at all.

nlin.CD↗

arXiv2006

Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.

nlin.SI↗

arXiv2003

The Darboux transformation and algebraic deformations of shape-invariant potentials

We investigate the backward Darboux transformations (addition of a lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, $m=0,1,2,...$, of deformations exists for each family of shape-invariant potentials. We prove that the $m$-th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules $\mathcal{P}^{(m)}_m\subset\mathcal{P}^{(m)}_{m+1}\subset...$, where $\mathcal{P}^{(m)}_n$ is a codimension $m$ subspace of $<1,z,...,z^n>$. In particular, we prove that the first ($m=1$) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules $\mathcal{P}^{(1)}_n = < 1,z^2,...,z^n>$. By construction, these algebraically deformed Hamiltonians do not have an $\mathfrak{sl}(2)$ hidden symmetry algebra structure.

quant-ph↗