Uniform approximation and explicit estimates for the prolate spheroidal wave functions

For fixed $c,$ Prolate Spheroidal Wave Functions (PSWFs), denoted by $\psi_{n, c},$ form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the $\psi_{n,c}$ in $n$ and $c,$ are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at $1$ in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs

Electronic and Molecular Dynamics by the Quantum Wave Packet Method

A solution to the time‐dependent Schrödinger equation is required in a variety of problems in physics and chemistry. In this chapter, recent developments of numerical and theoretical techniques for quantum wave packet methods efficiently describe the dynamics of molecular dynamics, and electronic dynamics induced by ultrashort laser pulses in atoms and molecules will be reviewed, particularly on the development of grid methods and time‐propagation or pseudo‐time evolution methods developed recently. Applications of the quantum wave packet for studying the reactive resonances in F + H2/HD and O + O2 reaction, dissociative chemisorption of water on transition‐metal surfaces, state‐to‐state reaction dynamics, state‐to‐state tetra‐atomic reaction dynamics using transition wave packet method and reactant coordinate method, and electronic dynamics in H2+ and H2 molecules will be presented