Boundary value problems in time for wave equations on \({\mathbb{R}}^ N\) (Q807853)

Let \(L_{\lambda}\) be the unbounded linear operator associated with the following boundary value problem (1), (2), (3.a) or (1), (2), (3.b): (1) \(L_{\lambda}w=w_{tt}-w_{rr}+[\lambda +(N-1)(N-3)/4r^ 2]w=h\), (t,r)\(\in (0,T)\times (R,\infty);\) (2) \(w(t,\cdot)\in L^ 2(R,\infty)\) a.e. \(t\in (0,T);\) (3.a) \(w(t+T,r)=w(t,r)\) a.e. (t,r)\(\in {\mathbb{R}}\times (R,\infty)\) or (3.b) \(\alpha_ 1w(0,r)-\alpha_ 2w_ t(0,t)=0\) and \(\beta_ 1w(T,r)+\beta_ 2w_ t(T,r)=0\), \(r\in (R,\infty)\) (with \(\alpha^ 2_ 1+\alpha^ 2_ 2\neq 0\), \(\beta^ 2_ 1+\beta^ 2_ 2\neq 0\), all nonnegative constants). The operator \(L_{\lambda}\) is associated with the radially symmetric form of the wave operator \(\partial_ t^ 2-\Delta +\lambda\), with given real number \(\lambda\in {\mathbb{R}}.\) Roughly speaking, the authors show that \(L_{\lambda}\) is a Fredholm operator with finite nonnegative index depending on \(\lambda\), considering the weighted Hilbert space \[ H_{\delta}^ m(D)=\{h\in H^ m(D)| \;e^{\delta r}h\in H^ m(D)\}\text{ for } D=(R,\infty)\text{ or } (0,T)\times (R,\infty)\text{ and } \delta >0. \] Their analysis bases on the separation of variables. Let \(\{\theta^ 2_ n\}\) and \(\{\xi_ n(t)\}\) be eigenvalues and eigenfunctions corresponding to the following problem: (4) \(\xi''+\theta^ 2\xi =0\), \(0<t<T,\) (5) \(\xi (0)=\xi (T)\), \(\xi'(0)=\xi '(T)\) or \(\alpha_ 1\xi (0)- \alpha_ 2\xi '(0)=0\), \(\beta_ 1\xi (T)+\beta_ 2\xi '(T)=0.\) Then, their theorem is the following. Theorem. If \(\theta_ j^ 2<\lambda \leq \theta^ 2_{j+1}\) for some j and \(0<\delta <(\lambda -\theta_ j^ 2)^{1/2}\) then the operator \(L_{\lambda}\) is a Fredholm operator onto \(L^ 2_{\delta}((0,T)\times (R,\infty))\) with index equal to the dimension of the linear space spanned by \(\{\xi_ n(t)\}_{n\leq j}\). Moreover, if the boundary conditions (5) are either periodic, Dirichlet, or Neumann, then there is a bounded partial inverse \[ K_{\lambda}: L^ 2_{\delta}((0,T)\times (R,\infty))\to H^ 1_{\delta}((0,T)\times (R,\infty)) \] such that \(L_{\lambda}w=h\) if and only if \(w=v+K_{\lambda}h\), where \(v\in Ker(L_{\lambda})\). In addition \(range(K_{\lambda})\subset Ker(L_{\lambda})^{\perp}\) with the orthogonality in \(H^ 1_{\delta}((0,T)\times (R,\infty))\).