Related papers: Associated Legendre Polynomials and Spherical Harm…
The authors prepared this booklet in order to make several useful topics from the theory of special functions, in particular the spherical harmonics and Legendre polynomials for any dimension, available to undergraduates studying physics or…
Recently published formulas for the surface and regular solid spherical harmonics and for the expansion of the product of two normalized associated Legendre functions with different centers in ellipsoidal coordinates (Telhat Ozdogan, Metin…
Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from…
Order parameters based on spherical harmonics and Fourier coefficients already play a significant role in condensed matter research in the context of systems of spherical or point particles. Here, we extend these types of order parameter to…
We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear…
Computing spherical harmonic decompositions is a ubiquitous technique that arises in a wide variety of disciplines and a large number of scientific codes. Because spherical harmonics are defined by integrals over spheres, however, one must…
Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate stable distributions under various conditions. A esult potentially applicable in a…
The angular momentum quantum number L of spherical harmonic Y_l_,_m based on an associated Legendre polynomial is nonnegative integer 0 1 2 ... and must never be a fraction. But the study in this paper found that the quantum number L…
We apply numerical optimization and linear algebra algorithms for classical computers to the problem of automatically synthesizing algorithms for quantum computers. Using our framework, we apply several common techniques from these…
The authors present SHarmonic, a new implementation of the spherical harmonics targeted for electronic-structure calculations. Their approach is to use explicit formulas for the harmonics written in terms of normalized Cartesian…
We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions $P_\nu^{-\mu}$ and $Q_\nu^{-\mu}$ of degrees $0 \leq \nu \leq 1,000,000$ and orders $-\nu \leq \mu \leq \nu$ on the interval…
In this paper, we report on very efficient algorithms for the spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library…
The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…
The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how…
Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same…
Quantum algorithms for scientific computing require modules implementing fundamental functions, such as the square root, the logarithm, and others. We require algorithms that have a well-controlled numerical error, that are uniformly…
We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier…
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for…
We propose a general method for constructing system-dependent basis functions for correlated quantum chemical calculations. Our construction combines features from several traditional approaches: plane waves, localized basis functions, and…