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 $D(L\sbsp{0}{m})$ = $D(L\sbsp{1}{m})$ / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_23828 |
| Date | January 1989 |
| Contributors | Pickett, Douglas Dean (Author), Goldstein, J. A (Thesis advisor) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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