FULL PAPER Complexity analysis of two families of orthogonal functions Piu Ghosh 1 | Debraj Nath 2 1 Department of Mathematics, Binapani Balika Vidyalaya (HS), North 24 Parganas, West Bengal, India 2 Department of Mathematics, Vivekananda College, Kolkata, West Bengal, India Correspondence Debraj Nath, Department of Mathematics, Vivekananda College, Kolkata, West Bengal, India. Email: debrajn@gmail.com Abstract We study the shape LMC (López-Ruiz, Mancini and Calvet), Fisher-Shannon (FS) and Cramér-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 Har- monic oscillator potentials whose solutions are given in terms of the classical orthog- onal polynomials. All these complexities are discussed in terms of the quantum number n and isospectrality parameter λ. KEYWORDS Cramér-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 [1–5] 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. [4,10–14] Another measure of complexity is the Fisher-Shannon (FS) complexity. [15–18] 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 quan- tum 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. [4,26–30] It is to be noted that various information theoretic measures of uncertainty and complexity have been studied in great detail for the well- known classical orthogonal polynomials which generate solutions of problems like the Harmonic oscillator, Coulomb potential, Morse potential, Pöschl-Teller potential, and so on. [31–38] However, there are not many exactly solvable potentials in quantum mechanics and consequently various formalisms for example, Darboux transformation, [39] Abraham-Moses method, [40] Mielnik procedure, [41] and so on have been developed to enlarge the class of exactly solvable potentials. A notable characteristic of these isospectral potentials is that their solutions are usually given in terms of orthogonal functions rather than classical orthogonal polynomials. Received: 2 February 2019 Revised: 11 April 2019 Accepted: 17 April 2019 DOI: 10.1002/qua.25964 Int J Quantum Chem. 2019;e25964. http://q-chem.org © 2019 Wiley Periodicals, Inc. 1 of 13 https://doi.org/10.1002/qua.25964