Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Abstract

A detailed study of the spinless (1+1)D free-particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter "flow" in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested. +∞ −∞ e −i p 2 2 m t e i px ψ 0 ( p)dp. (12) As we know, in analogy with (9), the evolution under the Schrödinger equation amounts in the p-domain to an energyrelated phase-factor, and hence to a simple quadratic-phase

Quasi‐Relativistic Heat Equation via Lévy Stable Distributions: Exact Solutions

“…We will use the technique analogical to that presented in. [12] Since Eq. (4) is linear in ∂ t its full solution is obtained by the action of the evolution operatorÛ(t) on the initial condition (i.c.)…”

“…For the extensive studies of the Salpeter equation see. [10][11][12] Analogously, the conventional diffusion equation [13] ∂ τ ψ(ξ, τ ) = D∂ 2 ξ ψ(ξ, τ )…”

Beyond the Light-Cone Propagation of Relativistic Wavefunctions: Numerical Results

“…Up to now, most works that have investigated this type of propagation, essentially in the context of the Salpeter equation, have used states that have tails at t = 0, such as initial Bessel functions [10] (one of the few cases for which analytical solutions can be obtained) or Gaussian wavepackets [8,[11][12][13]. If the states have initial compact support, we can meaningfully and numerically follow the fraction that remains inside the light cone as time evolves.…”