Let L be a linear, closed, densely defined in a Hilbert space operator,
not necessarily selfadjoint. Consider the corresponding wave equations
\begin{eqnarray} &(1) \quad \ddot{w}+ Lw=0, \quad
w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. \\ &(2)
\quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0,
\end{eqnarray}$\begin{array}{ccc}& \mathrm{\left(}\mathrm{1}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{\xa8}\\ \mathit{w}\end{array}\mathrm{+}\mathit{Lw}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{w}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{\u1e87}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{f,}\hspace{1em}\mathit{\u1e87}\mathrm{=}\frac{\mathit{dw}}{\mathit{dt}}\mathit{,}\hspace{1em}\mathit{f}\mathrm{\in}\mathit{H}\mathit{.}& \\ & \mathrm{\left(}\mathrm{2}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{\xa8}\\ \mathit{u}\end{array}\mathrm{+}\mathit{Lu}\mathrm{=}\mathit{f}{\mathit{e}}^{\mathrm{-}\mathit{ikt}}\mathit{,}\hspace{1em}\mathit{u}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{u\u0307}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}& \end{array}$
where k > 0 is a constant. Necessary and sufficient conditions are
given for the operator L not to have eigenvalues in the half-plane
Rez < 0 and not to have a positive eigenvalue at a given point kd2 > 0. These conditions are given in terms of the large-time
behavior of the solutions to problem (1) for generic f.
Sufficient conditions are given for the validity of a version of the limiting amplitude
principle for the operator L.
A relation between the limiting amplitude principle and the limiting absorption principle
is established.