On a diffusion on finite adeles and the Feynman-Kac integral

Let K be an algebraic number field. With K, we associate the ring of finite adeles AK. Following a recent result of Weisbart on diffusions on finite rational adeles AQ, we define the Vladimirov operator ΔAK on AK and define the Brownian motion on the group AK. We also consider the Schrödinger operat...

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Bibliographic Details
Published in:Journal of mathematical physics 2022-12, Vol.63 (12)
Main Author: Urban, Roman
Format: Article
Language:English
Subjects:
Brownian motion
Continuity (mathematics)
Number theory
Operators (mathematics)
Physics
Semigroups
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Summary:Let K be an algebraic number field. With K, we associate the ring of finite adeles AK. Following a recent result of Weisbart on diffusions on finite rational adeles AQ, we define the Vladimirov operator ΔAK on AK and define the Brownian motion on the group AK. We also consider the Schrödinger operator −HAK=−ΔAK+V with a potential operator V given by a non-negative continuous function v on AK. We prove a version of the Feynman–Kac formula for the Schrödinger semigroup generated by −HAK.
ISSN:0022-2488
1089-7658
DOI:10.1063/5.0111423