Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $.)


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34G05, 47D05

Retrieve articles in all journals with MSC: 34G05, 47D05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0410016-8
PII: S 0002-9947(1976)0410016-8
Keywords: Equipartition of energy, unitary group, hyperbolic equation, virial theorem
Article copyright: © Copyright 1976 American Mathematical Society