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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic equipartition of energy for differential equations in Hilbert space

Authors: Jerome A. Goldstein and James T. Sandefur
Journal: Trans. Amer. Math. Soc. 219 (1976), 397-406
MSC: Primary 34G05; Secondary 47D05
MathSciNet review: 0410016
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Abstract: Of concern are second order differential equations of the form $ (d/dt - i{f_1}(A))(d/dt - i{f_2}(A))u = 0$. Here A is a selfadjoint operator and $ {f_1},{f_2}$ are real-valued Borel functions on the spectrum of A. The Cauchy problem for this equation is governed by a certain one parameter group of unitary operators. This group allows one to define the energy of a solution; this energy depends on the initial data but not on the time t. The energy is broken into two parts, kinetic energy $ K(t)$ and potential energy $ P(t)$, and conditions on A, $ {f_1},{f_2}$ are given to insure asymptotic equipartition of energy: $ {\lim _{t \to \pm \infty }}K(t) = {\lim _{t \to \pm \infty }}P(t)$ for all choices of initial data. These results generalize the corresponding results of Goldstein for the abstract wave equation $ {d^2}u/d{t^2} + {A^2}u = 0$. (In this case, $ {f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda $.)

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Keywords: Equipartition of energy, unitary group, hyperbolic equation, virial theorem
Article copyright: © Copyright 1976 American Mathematical Society

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