{"title":"Adiabatic Klein–Gordon dynamics for the renormalized Nelson model","_id":"23153","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","supplementarymaterial":"yes","fulldoi":"https://doi.org/10.1007/s00526-026-03436-9","department":[{"_id":"RoSe"}],"arxiv":1,"publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"day":"01","issue":"10","doi":"10.1007/s00526-026-03436-9","oa":1,"researchdata_availability":"no","mathsc":["81Q05"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"file":[{"file_size":905142,"file_name":"2026_CalculusVariations_Brooks.pdf","creator":"dernst","checksum":"406bc9d144c35c9485d277c842d38c36","content_type":"application/pdf","date_created":"2026-10-19T07:53:14Z","file_id":"23166","success":1,"relation":"main_file","date_updated":"2026-10-19T07:53:14Z","access_level":"open_access"}],"volume":65,"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","das_tickbox":"1","external_id":{"pmid":["42865432"],"arxiv":["2509.14226"]},"article_type":"original","language":[{"iso":"eng"}],"abstract":[{"text":"We study the renormalized Nelson model in a semiclassical regime where the field becomes\r\nclassical while the particle remains quantum. The degree of classicality is measured by a\r\nsmall parameter ε << 1. In this scaling the particle evolves on microscopic times, whereas\r\nthe field exhibits nontrivial dynamics only on the macroscopic scale t = O(ε^−2). The natural\r\nsemiclassical model is the coupled Schrödinger–Klein–Gordon (SKG) system, which\r\nencodes the time-scale separation through an explicit ε-dependence. Based on this scale separation\r\nin SKG, we apply the adiabatic principle to derive a new PDE for the classical field,\r\nthe ε-free adiabatic Klein–Gordon (aKG) equation, where the field is driven by the instantaneous\r\nground state of the particle. Our main result is a norm approximation of the Nelson\r\ndynamics by the aKGsolution corrected by a quasi-free fluctuation dynamics around the classical\r\nfield, generated by a renormalized Bogoliubov–Nelson Hamiltonian. As a corollary, we\r\nobtain convergence of the reduced one-body densities for both subsystems, where the fluctuation\r\ncorrection vanishes, thereby justifying aKG as a semiclassical Born–Oppenheimer\r\ntype approximation of the renormalized Nelson model.","lang":"eng"}],"OA_place":"publisher","type":"journal_article","status":"public","quality_controlled":"1","PlanS_conform":"1","OA_type":"hybrid","scopus_import":"1","dataavailabilitystatement":"No data are associated with this article.","date_updated":"2026-10-19T07:55:13Z","author":[{"orcid":"0000-0002-6249-0928","full_name":"Brooks, Morris","id":"B7ECF9FC-AA38-11E9-AC9A-0930E6697425","last_name":"Brooks","first_name":"Morris"},{"full_name":"Mitrouskas, David Johannes","id":"cbddacee-2b11-11eb-a02e-a2e14d04e52d","last_name":"Mitrouskas","first_name":"David Johannes"}],"file_date_updated":"2026-10-19T07:53:14Z","intvolume":"        65","acknowledgement":"Morris Brooks gratefully acknowledges support from the ERC Advanced Grant CLaQS, grant agreement No. 834782. Open access funding provided by University of Zurich.","article_number":"270","publication_status":"published","year":"2026","date_created":"2026-10-18T22:01:52Z","publication":"Calculus of Variations and Partial Differential Equations","oa_version":"Published Version","citation":{"ama":"Brooks M, Mitrouskas DJ. Adiabatic Klein–Gordon dynamics for the renormalized Nelson model. <i>Calculus of Variations and Partial Differential Equations</i>. 2026;65(10). doi:<a href=\"https://doi.org/10.1007/s00526-026-03436-9\">10.1007/s00526-026-03436-9</a>","ista":"Brooks M, Mitrouskas DJ. 2026. Adiabatic Klein–Gordon dynamics for the renormalized Nelson model. Calculus of Variations and Partial Differential Equations. 65(10), 270.","ieee":"M. Brooks and D. J. Mitrouskas, “Adiabatic Klein–Gordon dynamics for the renormalized Nelson model,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 10. Springer, 2026.","apa":"Brooks, M., &#38; Mitrouskas, D. J. (2026). Adiabatic Klein–Gordon dynamics for the renormalized Nelson model. <i>Calculus of Variations and Partial Differential Equations</i>. Springer. <a href=\"https://doi.org/10.1007/s00526-026-03436-9\">https://doi.org/10.1007/s00526-026-03436-9</a>","mla":"Brooks, Morris, and David Johannes Mitrouskas. “Adiabatic Klein–Gordon Dynamics for the Renormalized Nelson Model.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 10, 270, Springer, 2026, doi:<a href=\"https://doi.org/10.1007/s00526-026-03436-9\">10.1007/s00526-026-03436-9</a>.","chicago":"Brooks, Morris, and David Johannes Mitrouskas. “Adiabatic Klein–Gordon Dynamics for the Renormalized Nelson Model.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer, 2026. <a href=\"https://doi.org/10.1007/s00526-026-03436-9\">https://doi.org/10.1007/s00526-026-03436-9</a>.","short":"M. Brooks, D.J. Mitrouskas, Calculus of Variations and Partial Differential Equations 65 (2026)."},"date_published":"2026-10-01T00:00:00Z","pmid":1,"publisher":"Springer","ddc":["500"],"month":"10"}