Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method

Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory

“…The identity element for the * -product is the Dirac distribution, denoted 1 * ≡ δ(z − z 0 ), an observation which we here accept without proof as it would require presenting the full theory of the * -product [30]. Similarly we accept without proof that for any bounded function f (z, z 0 ) of two variables, f * 0 = 1 * , while f * 1 = f and f * n+1 = f * f * n = f * n * f [80].…”

“…The central mathematical concept enabling the path-sum formulation of path-ordered exponentials is the * -product. This product is defined on a large class of distributions [30], however for the present work only its definition on smooth functions of two variables is required. For such functions the *product reduces to the Volterra composition, a product between functions first expounded by Volterra and Pérès in the 1920s [80] and which had largely fallen out of use by the early 1950s for a reason that appears, restrospectively, to be the lack of a mathematical theory of distributions.…”

Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory

“…As the size of the Hamiltonian increases, we expect the difference of computation times to grow in favor of codes evaluating the path-sum solutions, in particular, for codes exploiting path-sum's scale invariance property [16] or using Lanczos path-sum [24,27,28]. Lanczos path-sum is a type of pre-conditioning procedure for path-sum that finds a tridiagonal time-dependent matrix T ptq whose line 1 column 1 entry of the time-ordered exponential is the same as ing a truncation of T that is much smaller than U .…”

Computations of general Heun functions from their integral series representations

“…This product makes D into a non-commutative algebra [10,11] with unit element 1 * ≡ δ(z ′ −z). Now consider the ensemble Sm Θ D of distributions of D for which all smooth coefficients fi (z ′ , z) = 0, i.e.…”

“…which is the convolution-like product introduced by Volterra in his studies of integral equations, now known as the Volterra composition [12]. Not only do distributions of Sm Θ have * -inverses for all z ′ , z ∈ C except at a countable number of isolated points [11] but, most importantly for our purposes here, they have * -resolvents. Such resolvents, denoted (1 * − f ) * −1 are given by the Neumann series,…”