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Operator Random Walks and Quantum Oscillator
Lobachevskii Journal of Mathematics Pub Date : 2020-07-29 , DOI: 10.1134/s1995080220040186 Yu. N. Orlov , V. Zh. Sakbaev , D. V. Zavadsky
中文翻译:
操作员随机游走和量子振荡器
更新日期:2020-07-29
Lobachevskii Journal of Mathematics Pub Date : 2020-07-29 , DOI: 10.1134/s1995080220040186 Yu. N. Orlov , V. Zh. Sakbaev , D. V. Zavadsky
Abstract
We investigate asymptotic properties of a sequence whose elements are probability distributions which correspond to compositions of independent random operators. We prove the following statements:1) a sequence whose elements are the expected values which corresponds to compositions of independent identically distributed random semigroups converges to a semigroup which is generated by the expected value of the random generator;2) an analog of the law of large numbers for a sequence whose elements are compositions of independent identically distributed random semigroups;3) a sequence whose elements are compositions of random shift operators on position space converges to a semigroup generated by the Laplace operator;4) a sequence whose elements are compositions of random shift operators on position space and random shift operators in momentum space converges to a semigroup generated by the Hamiltonian of the quantum oscillator;5) a sequence whose elements are compositions of random shift operators on phase space converges to a semigroup generated by the Hamiltonian of the quantum oscillator.中文翻译:
操作员随机游走和量子振荡器