The author of this interesting paper considers the reduced (in the angular coordinate \(\varphi \)) wave equation and Klein-Gordon equation on a Kerr background and in the framework of \(C^0\)-semigroup theory. The Kerr metric in Boyer-Lindquist coordinates \(t,r,\theta , \varphi \) has the form \[ g=(1-2Mr/\Sigma) dr^2+(4Mar\sin^2\theta /\Sigma) dt d\varphi -(\Sigma /\triangle) dr^2-\Sigma d\theta^2- \]
\[ -(r^2+a^2+2Ma^2r\sin^2\theta /\Sigma)\sin\theta d\varphi^2, \] where \(M\) is the mass, \(a\in [0,M]\) is the rotational parameter, \(\triangle\equiv r^2-2Mr+a^2\), \(\Sigma\equiv r^2+a^2\cos^2\theta \). Here \(-\infty <t<+\infty \), \(r_+<r<+\infty \), \(-\pi <\varphi <\pi \), \(0<\theta <\pi \) (\(r_+\equiv M+\sqrt {M^2-a^2}\)). For each equation it is shown that the initial value problem is well-posed. This means that there exists a unique solution which depends continuously on the data. It is shown that the spectrum of the semigroup’s generator coincides with the spectrum of an operator polynomial whose coefficients can be read off from the equation. In this way, the problem of deciding stability is reduced to a spectral problem. For the wave equation, it is shown that the resolvent of the semigroup’s generator and the corresponding Green’s functions can be computed using spheroidal functions. It is to be expected that, analogous to the case of a Schwarzschild background, the quasinormal frequencies of the Kerr black hole appear as resonances. The stability of the solutions of the reduced Klein-Gordon equation is proven for sufficiently large masses.