{"title":"Self-Adjointness of a Class of Multi-Spin–Boson Models with Ultraviolet Divergences","authors":"Davide Lonigro","doi":"10.1007/s11040-023-09457-6","DOIUrl":null,"url":null,"abstract":"<div><p>We study a class of quantum Hamiltonian models describing a family of <i>N</i> two-level systems (spins) coupled with a structured boson field of positive mass, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences. Spin–spin interactions which do not modify the total number of excitations are also included. Generalizing previous results in the single-spin case, we provide explicit expressions for the self-adjointness domain and the resolvent of these models, both of them carrying an intricate dependence on the spin–field and spin–spin coupling via a family of concatenated propagators. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors via normalizable ones, for example an ultraviolet cutoff.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09457-6.pdf","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-023-09457-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
We study a class of quantum Hamiltonian models describing a family of N two-level systems (spins) coupled with a structured boson field of positive mass, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences. Spin–spin interactions which do not modify the total number of excitations are also included. Generalizing previous results in the single-spin case, we provide explicit expressions for the self-adjointness domain and the resolvent of these models, both of them carrying an intricate dependence on the spin–field and spin–spin coupling via a family of concatenated propagators. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors via normalizable ones, for example an ultraviolet cutoff.
期刊介绍：
MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas.
The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process.
The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed.
The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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