Time-Dependent Logarithmic Perturbation Theory for Quantum Dynamics: Formulation and Applications

Quantum Physics [Submitted on 16 Apr 2026] Time-Dependent Logarithmic Perturbation Theory for Quantum Dynamics: Formulation and Applications Juan Carlos del Valle, Paul Bergold, Karolina Kropielnicka We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schrödinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime. Comments: First version, 31 pages, 4 figures Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Analysis of PDEs (math.AP) Cite as: arXiv:2604.14812 [quant-ph]   (or arXiv:2604.14812v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.14812 Focus to learn more Submission history From: Juan Carlos Del Valle Rosales PhD [view email] [v1] Thu, 16 Apr 2026 09:33:02 UTC (269 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: math math-ph math.AP math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)