  The tidal response of compact objects in an inspiraling binary system is
measured by a set of tidal Love and dissipation numbers imprinted in the
gravitational waveforms. While a four-dimensional black hole in vacuum within
General Relativity has vanishing Love numbers, a black hole in alternative
theories of gravity can acquire non-vanishing Love numbers. The dissipation
numbers may quantify Planckian corrections at the horizon scale. These
properties will allow a test of classical theories of gravity in the
strong-field regime with gravitational-wave observation. Since black holes are
not in the exact vacuum environment in astrophysical situations, the following
question arises: can the environment affect the tidal response? In this paper,
we investigate the stability of the tidal response of a Schwarzschild black
hole for frequency-dependent tidal-field perturbations against a small
modification of the background. Our analysis relies on the scattering theory,
which overcomes difficulties in computing the relativistic tidal Love numbers.
The tidal Love and dissipation numbers can be understood from the property of
sufficiently low-frequency scattering waves. We show that the tidal Love
numbers are sensitive to the property of the modification. Therefore, we need
careful consideration of the environment around the black hole in assessing the
deviation of the underlying theory of gravity from General Relativity with the
Love numbers. The modification has less impact on the dissipation numbers,
indicating that quantifying the existence of the event horizon with them is not
spoiled. We also demonstrate that in a composite system, i.e., a compact object
with environmental effects, the Love and dissipation numbers are approximately
determined by the sum of the numbers of each component.
