The leading adiabatic response of a static black hole to a slowly varying external scalar exists and is unique exactly when a single matching condition on the kernel generator is nonzero.
Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin (1995)
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Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability
The leading adiabatic response of a static black hole to a slowly varying external scalar exists and is unique exactly when a single matching condition on the kernel generator is nonzero.