  We present a theoretical framework for linking quasar properties, such as
quasar age, to the surrounding Ly$\alpha$ emission intensity. In particular, we
focus on a method for mapping the large-scale structure of Ly$\alpha$ emission
intensity with galaxy spectra from wide-field spectroscopic surveys, e.g., the
Subaru Prime Focus Spectrograph (PFS) or the Dark Energy Spectroscopic
Instrument (DESI), and consider the quasar-induced Ly$\alpha$ emission from the
intergalactic medium (IGM). To do this, we construct a theoretical model based
on two physical processes: resonant scattering of quasar Ly$\alpha$ photons and
fluorescence due to quasar ionizing photons, finding that the fluorescence
contribution due to optically thick gas clouds is dominant. Taking into account
the light cone effect and assuming a typical quasar spectrum, we calculate the
fluorescence contribution to the spectrum stacked within each bin of the
separation angle from the quasar as a function of quasar age. Furthermore, we
compute the quasar-Ly$\alpha$ emission cross-correlation and its SNR for the
planned PFS survey. The predicted signal can account for $\sim10\%$ of the
measurements indicated from the BOSS and eBOSS surveys in the outer region of
$>10\ \rm{cMpc}\ \rm{h}^{-1}$. The predicted SNR is not enough to detect the
quasar-induced contribution, while it is enhanced by including contributions
from other Ly$\alpha$ emission sources, e.g., star-forming galaxies. We discuss
other possible contributions to the Ly$\alpha$ emission excess around quasars,
the efficiency of using spectroscopic fibers, and the redshift dependence of
our model.

Astrophysics of Galaxies