Quasinormal mode

Remarks

The boundary conditions say that the wave only leaves the region: outward to $x = +\infty$ and inward to $x = -\infty$. Then $u$ satisfies $u'' + (\omega^2 - V) u = 0$, and $\omega$ is determined by the boundary conditions as in an eigenvalue problem. If $\operatorname{Im} \omega < 0$ the mode decays in time, but $u$ grows exponentially as $x \to \pm\infty$, so it is not square integrable and $\omega$ is not an eigenvalue of a self-adjoint operator. For this reason quasinormal frequencies are often defined as poles of the meromorphic continuation of the resolvent, also called resonances.

Example: perturbations of the Schwarzschild black hole of mass $M$ reduce, mode by mode in spherical harmonics, to this equation. The variable $x$ is the tortoise coordinate, with the horizon at $x = -\infty$, and $V$ is the Regge–Wheeler or Zerilli potential. The least damped gravitational mode with angular number $\ell = 2$ has $M\omega \approx 0.3737 - 0.0890\, i$ in units $G = c = 1$.

Non-example: a string of length $\pi$ with fixed ends obeys $\partial_t^2 \psi = \partial_x^2 \psi$ with $u(0) = u(\pi) = 0$. Its modes $\sin(nx)\, e^{-int}$ have real frequencies and do not decay; they are normal modes, because no energy leaves the string, and the outgoing conditions are not satisfied.