An introduction to analysis of Rényi complexity ratio of quantum states for central potential

Abstract

Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for six diatomic molecules (CO, NO, N2, CH, H2, and ScH) and for other quantum systems.

Algebraic $\mathcal{L}_{q}$-norms and complexity-like properties of Jacobi polynomials-Degree and parameter asymptotics

“…First, the extension of these results to the varying Jacobi polynomials [70][71][72] (i.e., when the parameters depend on the polynomial degree) as well as to the exceptional Jacobi polynomials [42,73,74], which are very useful to standard and supersymmetric quantum mechanics [41]. Second, the determination of the general statistical complexity measures [75] of Fisher-Rényi [25,28,31,32,77] and LMC-Rényi [24,26,28,52,76] types for the standard and varying Jacobi polynomials; this includes the calculation of the Rényi entropy of such polynomials. These open issues are not only interesting per se but also because of their chemical and physical applications, especially for the extreme quantum states of highly excited Rydberg and high dimensional types of numerous atomic and molecular systems whose bound states are described by wavefunctions controlled by these polynomials.…”

“…Recently the entropy-and complexity-like properties of these polynomials, which determine their spreading on the orthogonality interval, have begun to be investigated by means of the entropy-and complexity-like measures [11,48,49] of the associated Rakhmanov density ρ n (x) = p 2 n (x) h(x). This normalized-to-unity probability density function governs the (n → +∞)-asymptotics of the ratio of two polynomials with consecutive orders [57], and characterizes the Born's probability density of the bound stationary states of a great deal of quantum-mechanical potentials which model numerous atomic and molecular systems [12,14,41,[50][51][52]. The numerical evaluation of the integral functionals corresponding to the dispersion, entropic and complexity measures of the HOPs by means of the standard quadratures is not convenient, because the highly oscillatory nature of the integrand renders Gaussian quadrature ineffective as the number of quadrature points grows linearly with n and the evaluation of high-degree polynomials are subject to round-off errors.…”