Time-independent perturbation theory for quasinormal-mode solutions in quantum mechanics

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Abstract
Quasinormal modes are characterized by complex energy eigenvalues and by eigenfunctions that are outgoing waves at infinity, and are therefore not normalizable in the usual sense. In the space of such wave functions, the operators of quantum mechanics are not Hermitian, and the familiar tools break down. We reformulate the time-independent perturbation theory for quasinormal modes and express the corrections to the complex energy in terms of matrix elements. One main feature of the results is the appearance of an alternative definition for the norm, which is finite for quasinormal modes. Explicit formulas are given for the real and imaginary parts of the first-order correction and for the imaginary part of the second-order correction, the latter in the approximation where the original quasinormal mode is extremely narrow. This restricted (but physically most useful) part of the perturbative formalism depends only on the first-order matrix elements, and the simple form of the second-order correction can be interpreted in terms of a generalized golden rule.