Complexity analysis of two families of orthogonal functions

Abstract

We study the shape LMC (López-Ruiz, Mancini and Calvet), Fisher-Shannon (FS) andCramér-Rao (CR) complexities of two families of orthogonal functions associated with the solutions of isospectral deformations of the Pöschl-Teller and Harmonic oscillator potentials. We have compared the behavior of these complexities for the orthogonal functions with the complexities associated with Pöschl-Teller and Harmonic oscillator potentials whose solutions are given in terms of the classical orthogonal polynomials. All these complexities are discussed in terms of the quantum number n and isospectrality parameter λ. K E Y W O R D SCramér-Rao complexity, Fisher-Shannon complexity, LMC complexity 1 | INTRODUCTION Application of statistical measures in physical as well as in social sciences has a role of growing importance. There exists a vast literature on the use of various measures of complexity in different contexts for example, dynamical systems, cellular automata, neural networks, social sciences, complex molecules, geophysical and astrophysical processes etc. In particular, López-Ruiz, Mancini, and Calvet (LMC) complexity [1][2][3][4][5] has been computed in position and momentum spaces [6] for the density functions of the hydrogen-like atoms and the isotropic harmonic oscillator. [7,8] The LMC complexity is defined as a product of two factors-one of which is a measure of the disequilibrium, that is, it quantifies the departure of the probability density from uniformity while the other one is the Shannon entropy which is a measure of uncertainty or randomness. [9] The modified LMC complexity, that is, the shape LMC is the product of the Shannon length and the disequilibrium [2,4,5] and have been studied in different contexts. [4,[10][11][12][13][14] Another measure of complexity is the Fisher-Shannon (FS) complexity. [15][16][17][18] It is defined as the product of the Fisher information [19] and Shannon entropic power. The LMC and the FS complexity measures have been applied in different fields of physics such as multi electron systems in position and momentum spaces, [20,21] analysis of signals, [15] electron correlation, [16] atomic systems and ionization processes [18,22] and in quantum mechanics. [7,8,23] The third complexity measure that we shall study is the Cramér-Rao (CR) complexity which is defined as the product of the Fisher information and the variance of the density function measuring the degree of deviation from the mean value. [20,22,24,25] The three complexities mentioned above share a set of characteristics, namely, they are (a) dimensionless (b) bounded below by unity, and (c) minimum for two extreme distributions which correspond to perfect order and maximum disorder. Also they are invariant under replication, translation and scaling transformation. [4,[26][27][28][29][30] It is to be noted that various information theoretic measures of uncertainty and complexity have been studied in great detail for the wellknown classical orthogonal polynomials which generate solutions of problems like the Harm...

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Thermal Uncertainty Relations and LMC Structural Quantifiers

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