Scattering theory for higherorder equations

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Title 
Scattering theory for higherorder equations 
Author 
Pickett, Douglas Dean 
School 
Tulane University 
Academic Field 
Mathematics 
Abstract 
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, skewadjoint 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 skewadjoint 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, selfadjoint 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 $D(L\sbsp{0}{m})$ = $D(L\sbsp{1}{m})$ 
Language 
eng 
Advisor(s) 
Goldstein, J. A 
Degree Date 
1989 
Degree 
Ph.D 
Publisher 
Tulane University 
Publication Date 
1989 
Source 
Source: 44 p., Dissertation Abstracts International, Volume: 5011, Section: B, page: 5108 
Identifier 
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Rights 
Copyright is in accordance with U.S. Copyright law 
Contact Information 
acase@tulane.edu 
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