# Scattering theory for higher-order equations

## Description

Let $\cal{H}$ be a Hilbert space. Of concern is the scattering problem for pairs of abstract hyperbolic equations of two particular forms. In each case, we study the problem by reducing the equations to systems on associated Hilbert spaces First we consider the equations$$\prod\limits\sbsp{i=1}{2\sp N}\left({d\over dt}-B\sbsp{i}{k}\right) u\sb{k}(t),\qquad t\in {\bf R}\leqno\rm(k)$$where for $k$ = 0 and $k$ = 1, $\{B\sbsp{i}{k}\}\sbsp{i = 1}{2\sp N}$ is a family of commuting, skew-adjoint operators on $\cal H$, with $B\sbsp{i}{k} - B\sbsp{j}{k}$ injective when $i \ne j$. For $k$ = 0, 1, equation (k) reduces to$${dv\sb{k}\over dt}={\cal A}\sb{k}v\sb{k}(t),\quad t\in{\bf R}, \quad v\sb{k}(t)\in {\cal H}\sp{2\sp N},\leqno\rm(k\prime)$$where ${\cal A}\sb{k}$ is a skew-adjoint matrix of operators on ${\cal H}\sp{2\sp N}$. Results are obtained for the scattering problem related to systems (0$\prime$) and (1$\prime$) Now let $L$ be a positive, self-adjoint operator on $\cal H$, and consider the equation$$\prod\limits\sbsp{j=1}{m}(\partial\sbsp{t}{2} + \alpha\sb{j}L)u(t)=0,\qquad t\in {\bf R}$$where the $\alpha\sb{j}$ are distinct positive constants. Results are obtained for the scattering problem related to two such equations, that is, for $L$ = $L\sb{0}$ and $L$ = $L\sb1$. In particular, we obtain results for the case where $D(L\sbsp{0}{m})$ = $D(L\sbsp{1}{m})$