  In this study, we investigate the pseudospectrum and spectrum (in)stability
of quantum corrected Schwarzschild black hole. Methodologically, we use the
hyperboloidal framework to cast the quasinormal mode (QNM) problem into an
eigenvalue problem associated with a non-selfadjoint operator, and then the
spectrum and pseudospectrum are depicted. Besides, the invariant subspace
method is exploited to improve the computational efficiency for pseudospectrum.
The investigation into the spectrum (in)stability entails two main aspects. On
the one hand, we calculate the spectra of the quantum corrected black hole,
then by the means of the migration ratio, the impact of the quantum correction
effect on the Schwarzschild black hole has been studied. The results indicate
that the so-called ``migration ratio instability" will occur for small black
holes with small angular momentum number l. In the eikonal limit, the migration
ratios remain the same for each overtone. On the other hand, we study the
spectrum (in)stability of the quantum corrected black hole by directly adding
some particular perturbations into the effective potential, where perturbations
are located at the event horizon and null infinity, respectively. There are two
interesting observations under the same perturbation energy norm. First,
perturbations at infinity are more capable of generating spectrum instability
than those at the event horizon. Second, we find that the peak distribution can
lead to the instability of QNM spectrum more efficiently than the average
distribution.
