  Upcoming surveys of the large-scale structure of our Universe will employ a
large coverage area of about half of the sky and will significantly increase
the observational depth. With these surveys, we will be able to cross-correlate
CMB gravitational lensing and galaxy surveys divided into narrow redshift bins
to map the evolution of the cosmological parameters with redshift. In this
paper, we study the effect of the redshift bin mismatch of objects that is due
to photometric redshift errors in tomographic cross-correlation measurements.
We used the code FLASK to create Monte Carlo simulations of the Vera C. Rubin
Observatory Legacy Survey of Space and Time (LSST) and Planck CMB lensing
convergence. We simulated log-normal fields and divided galaxies into nine
redshift bins with the Gaussian and modified Lorentzian photometric redshift
errors. To estimate the parameters, we used angular power spectra of CMB
lensing and galaxy density contrast fields and the maximum likelihood
estimation method. We show that even with simple Gaussian errors with a
standard deviation of $\sigma(z)=0.02(1+z)$, the galaxy auto-power spectra in
tomographic bins are offset by $2-15\%$. The estimated cross-power spectra
between galaxy clustering and CMB lensing are also biased, with smaller
deviations $<5\%$. As a result, the $\sigma_{8}$ parameter deviates between
$0.2-1.2\,\sigma$ due to the redshift bin mismatch of the objects. We propose a
computationally fast and robust method based on the scattering matrix approach
of Zhang et al. (2010), to correct for the redshift bin mismatch of the
objects. The estimates of the parameters in tomographic studies such as the
linear galaxy bias, the cross-correlation amplitude, and $\sigma_{8}$ are
biased due to the redshift bin mismatch of the objects. The biases in these
parameters are alleviated with our scattering matrix approach.
