Asymptotic equipartition of energy for differential equations in hilbert space


Of concern are second order differential equations of the form (d/dt – if1(A))u= 0. Here A is a selfadjoint operator and f1, f2 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, f1, f2we given to insure asymptotic equipartition of energy: lim^^ A(f) = lim^t+eo P{t) for all choices of initial data. These results generalize the corre-˜ 2 2 2 sponding results of Goldstein for the abstract wave equation d ufdt + A u = 0. (In this case, f1λ= λ, f2(λ)=-λ. © 1976, American Mathematical Society.

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Transactions of the American Mathematical Society