Abstract equipartition of energy theorems
Let H be a self-adjoint operator on a complex Hilbert space H. The solution of the abstract Schrödinger equation idu dt = Hu is given by u(t) = exp(-itH)u(0). The energy E = ∥u(t)∥2 is independent of t. When does the energy break up into different kinds of energy E = ∑j = 1N Ej(t) which become asymptotically equipartitioned ? (That is, Ej(t) → E N as t → ± ∞ for all j and all data u(0).) The "classical" case is the abstract wave equation d2v dt2 + A2v = 0 with A self-adjoint on H1. This becomes a Schrödinger equation in a Hilbert space H (essentially H is two copies of H1), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C*-algebra approach to quantum field theory and statistical mechanics. (2) Let A1,..., AN be commuting self-adjoint operators with N = 2 or 4. Then the equation Πj = 1N ( d dt - iAj) u(t) = 0 admits equipartition of energy if and only if exp(it(Aj - Ak)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k. © 1979.
Journal of Mathematical Analysis and Applications
Goldstein, J., & Sandefur, J. (1979). Abstract equipartition of energy theorems. Journal of Mathematical Analysis and Applications, 67 (1), 58-74. https://doi.org/10.1016/0022-247X(79)90006-4