Abstract equipartition of energy theorems

AbstractLet H be a self-adjoint operator on a complex Hilbert space H. The solution of the abstract Schrödinger equation idudt = 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) → EN as t → ± ∞ for all j and all data u(0).) The “classical” case is the abstract wave equation d2vdt2 + 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 (ddt − 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