Comparing a gauge-invariant formulation and a "conventional complete gauge-fixing approach" for $l=0,1$ mode perturbations on the Schwarzschild background spacetime

Comparison of the gauge-invariant formulation for $l=0,1$-mode perturbations on the Schwarzschild background spacetime proposed in [K.~Nakamura, Class. Quantum Grav. {\bf 38} (2021), 145010.] and a conventional complete gauge-fixing approach'' in which we use the spherical harmonic functions $Y_{lm}$ as the scalar harmonics from the starting point is discussed. Although it is often said that gauge-invariant formulations in general-relativistic perturbations are equivalent to complete gauge-fixing approaches,‘’ as the result of this comparison, we conclude that the derived solutions through the proposed gauge-invariant formulation and those through a ``conventional complete gauge-fixing approach’’ are different. It is pointed out that there is a case where the boundary conditions and initial conditions are restricted in a conventional complete gauge-fixing approach.