General solution of the exceptional Hermite differential equation and its minimal surface representation
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Annales Henri Poincar´ e
General solution of the exceptional Hermite differential equation and its minimal surface representation V. Chalifour
and A. M. Grundland
Abstract. The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition λ = (1) and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the general analytic solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is complemented by non-polynomial solutions. We present an implementation of the obtained results for the surfaces expressed in terms of the general solution making use of the classical Enneper–Weierstrass formula for the immersion in the Euclidean space E3 , leading to minimal surfaces. Three-dimensional displays of these surfaces are presented. Mathematics Subject Classification. 33C45, 34B24, 53A10. Keywords. Exceptional orthogonal polynomial, Hermite polynomial, integrable system, minimal surface, Enneper–Weierstrass immersion formula.
1. Introduction Over the last decade, the problem of exceptional orthogonal polynomials (XOPs) has generated a great deal of interest and activity in several areas of mathematics and physics. Most of these activities focused on Jacobi and Laguerre XOPs (see e.g. [4,5,19,22,27,32,35,36,38] and references therein), Hermite XOPs
V. Chalifour, A. M. Grundland
Ann. Henri Poincar´e
(see e.g. [6,17,18,20,21,26]) and multi-indexed orthogonal polynomials (see e.g. [31,33] and references therein). Exceptional orthogonal polynomials have been shown to play an essential role in several branches of physics, mostly related to the quantum harmonic oscillator. In particular, Jacobi XOPs were applied to the description of the Kepler–Coulomb quantum model [24] and were seen as having an electrostatic interpretation [10]. Hermite XOPs appeared first in the context of a time-independent Schr¨ odinger problem [12,13,37]. They were also applied to the description of coherent states [23]. The subject of study in our paper is exceptional Hermite polynomials. A substantial progress has been made recently in this area [6,15,18–21,26,28], among others by G´ omez-Ullate et al. [17], who provided a general formula for exceptional Hermite differential operators with an arbitrary partition. However, the associated ordinary differential equation (ODE) is a very complex one with a high degree of freedom and its general solution has yet to be established. In our study, we focus on a specific family of complex Hermite XOPs of codimension two, defined by the fixed partition
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