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We present a new method for determining the frequencies and wavefunctions ofblack hole quasinormal modes (QNMs) and Regge poles. The key idea is a novelansatz for the wavefunction, which relates the high-$l$ wavefunctions to nullgeodesics which start at infinity and end in perpetual orbit on the photonsphere. Our ansatz leads naturally to the expansion of QNMs in inverse powersof angular momentum $L = l + 1/2$ (in 4D), and to the expansion of Regge polesin inverse powers of frequency. The expansions can be taken to high orders. Webegin by applying the method to the Schwarzschild spacetime, and validate ourresults against existing numerical and WKB methods. Next, we generalise themethod to treat static spherically-symmetric spacetimes of arbitrary spatialdimension. We confirm that, at lowest order, the real and imaginary componentsof the QNM frequency are related to the orbital frequency and the Lyapunovexponent for geodesics at the unstable orbit. We apply the method to fivespacetimes of current interest, and conclude with a discussion of theadvantages and limitations of the new approach, and its practical applications.

On an Expansion Method for Black Hole Quasinormal Modes and Regge Poles