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}![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20161010080505457-0193:S0973534813081169:S0973534813081169_eqnU6.png)
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.