Matrix-valued Schrödinger operators over finite adeles
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Published:2023-03-30
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Volume:
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ISSN:0219-0257
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Container-title:Infinite Dimensional Analysis, Quantum Probability and Related Topics
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language:en
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Short-container-title:Infin. Dimens. Anal. Quantum. Probab. Relat. Top.
Affiliation:
1. Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Abstract
Let [Formula: see text] be an algebraic number field. With [Formula: see text] we associate the ring of finite adeles [Formula: see text] In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group [Formula: see text] Specifically, we consider matrix-valued Hamiltonian operators [Formula: see text] where [Formula: see text] is the Vladimirov operator and [Formula: see text] is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text] This formula is given in terms of the ordered time exponentials.
Publisher
World Scientific Pub Co Pte Ltd
Subject
Applied Mathematics,Mathematical Physics,Statistics and Probability,Statistical and Nonlinear Physics