Scattering theory for higher order equations (Q805998)

Consider the skew-adjoint commutative operators \(B_ 0,B_ 1\) on a Hilbert space H and define skew-adjoint operators \(A_ 0=(B_ 0+B_ 1)^ c,\quad A_ 1=(B_ 0-B_ 1)^ c.\) Setting \(A_{(2)}=\begin{pmatrix} A_ 0&A_ 1 \\ A_ 1&A_ 0 \end{pmatrix}\) and \(B_{(2)}=\left( \begin{matrix} B_ 0\\ 0\end{matrix} \quad \begin{matrix} 0\\ B_ 1\end{matrix} \right)\) on \(H\oplus H\), we see that \(A_{(2)}\) is unitarily equivalent to \(B_{(2)}\) by \(A_{(2)}=U_ 2B_{(2)}U_ 2\), \(U_ 2=\frac{1}{\sqrt{2}}\begin{pmatrix} 1&1 \\ 1&-1 \end{pmatrix}\). In case that one has two families of commuting pairs \(\{B^ n_ k\}\), \(n=0,1\), the scattering theory for \[ (\frac{d}{dt}-B^ n_ 0)(\frac{d}{dt}-B^ n_ 1)u(t)=0,\quad n=0,1, \] may be reformulated in terms of \(A^ n_{(2)}\) and, hence, in terms of \(B^ n_{(2)}\), \(n=0,1\), on \(H^ 2=H\oplus H\). The author generalizes this example, to the case of the initial value problem for \[ \prod^{2^ N-1}_{j=0}(\frac{d}{dt}-B^ n_ j)u(t)=0,\quad n=0,1, \] where \(B^ n_ j\), \(j=0,1,...,2^ N-1\) are families of \(2^ N\) commuting skew-adjoint operators on a complex Hilbert space H and \(B^ n_ i-B^ n_ j\) is injective when \(i\neq j\). As in the example, this may be reduced to the consideration of the problem \[ (\frac{d}{dt}-A^ n_{(N)})u(t)=0,\quad t\in {\mathbb{R}}\text{ on } H^ 2. \] This follows by a simple inductive argument. As previously there exist constant coefficient (self-adjoint) unitary matrices \(U_ N\) such that \(A_{(N)}=U_ NB_{(N)}U_ N\), \(n=0,1.\) The principal result of the paper is perhaps now obvious, namely: The wave operators \(\Omega^ j_{\pm}=W_{\pm}(iB^ 1_ j,iB^ 0_ i)\) exist (and are complete) for \(0\leq j\leq 2^ N-1\) if and only if the wave operators \(W_{\pm}(iA^ 1_{(N)},iA^ 0_{(N)})\) exist (and are complete).