Asymptotic partition of energy for abstract uniformly propagative systems (Q2640770)

The operator A which is self-adjoint on the direct sum \({\mathcal H}=\oplus H\) of N copies of the Hilbert space H and has the matrix representation with the entries \(A_{jk}\) \((j,k=1,2,...,N)\) which are Borel functions of a family \(\{C_ k\}\) \((k=1,2,...,J)\) of selfadjoint and commuting operators in H, is a strongly uniformly propagative operator matrix if (i) there is a selfadjoint operator \(\Lambda_ j(C)\) associated with each eigenvalue \(\Lambda_ j(\lambda)\) of A(C) and (ii) the eigenprojection \(P_ j(\lambda)\) associated with \(\Lambda_ j(\lambda)\) has constant rank \(m_ j\) for each of the eigenvalues \(\Lambda_ j(\lambda).\) If M(C) is an arbitrary (N\(\times N)\) operator matrix generated by C and defined on all of \({\mathcal H}\), a weak solution \(U(t)=\exp (itA(C))U_ 0\) of the evolution equation \(\dot U=iAU\) with \(U_ 0\in {\mathcal H}\) satisfies the asymptotic property \[ \| M(c)U(t)\|^ 2=\sum^{K}_{j=1}\| M(C)P_ j(C)U_ 0\|^ 2+o(1) \] as \(t\to \pm \infty\), K being the total number of eigenvalues. If \(\pi_ j\) is the canonical \({\mathcal H}\)-orthogonal projection on the jth component of \({\mathcal H}\) \((j=1,2,...,N)\) and \(U_ j(t)=\pi_ jU(t)\), the asymptotic contribution of \(\| U_ j(t)\|^ 2\) to the total energy \(E=\| U_ 0\|^ 2\) as \(t\to \pm \infty\) is \(\phi_ j=\{\sum^{K}_{k=1}\| \pi_ jP_ k(C)U_ 0\|^ 2\}/E\) \((j=1,2,...,N)\) and \(\sum P_ j=1\). If the ratios \(\rho_ j\) are independent of the initial value \(U_ 0\in {\mathcal H}-\{0\}\), then \(\rho_ j=N^{-1}\) for each of the N values of j and thus the paper proves the property of equi-partition of energy for the abstract system.