A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus

Abstract

We obtain a basis- and integral-free representation of the $n$-term in time-dependent perturbation theory of the quantum time-evolution operator. The main technical tool to construct the aforementioned representation comprises the Omega matrix calculus; that is, an extension of MacMahon’s partition analysis to the realm of matrix calculus. In particular, we show that if we specialize our formulation to the time-independent and finite-dimensional case, we recover Putzer (1966) original formulation to compute the matrix exponential avoiding the Jordan canonical form. Furthermore, if we use an explicit basis for the time-independent part of the generator of evolution, then our work implies a representation of the perturbation expansion similar to an approach discussed in the recent work of Kalev and Hen (2021) using divided differences. Finally, we obtain closed form expressions which generalize and unify all the perturbation calculations considered in Kalev and Hen (2021) and advance some considerations regarding degeneracy associated with the unperturbed Hamiltonian in the context of adiabatic switching.