  We analyze the Schr\"odingerisation method for quantum simulation of a
general class of non-unitary dynamics with inhomogeneous source terms. The
Schr\"odingerisation technique, introduced in \cite{JLY22a,JLY23}, transforms
any linear ordinary and partial differential equations with non-unitary
dynamics into a system under unitary dynamics via a warped phase transition
that maps the equations into a higher dimension, making them suitable for
quantum simulation. This technique can also be applied to these equations with
inhomogeneous terms modeling source or forcing terms or boundary and interface
conditions, and discrete dynamical systems such as iterative methods in
numerical linear algebra, through extra equations in the system. Difficulty
airses with the presense of inhomogeneous terms since it can change the
stability of the original system.
  In this paper, we systematically study--both theoretically and
numerically--the important issue of recovering the original variables from the
Schr\"odingerized equations, even when the evolution operator contains unstable
modes. We show that even with unstable modes, one can still construct a stable
scheme, yet to recover the original variable one needs to use suitable data in
the extended space. We analyze and compare both the discrete and continuous
Fourier transforms used in the extended dimension, and derive corresponding
error estimates, which allows one to use the more appropriate transform for
specific equations. We also provide a smoother initialization for the
Schrod\"odingerized system to gain higher order accuracy in the extended space.
We homogenize the inhomogeneous terms with a stretch transformation, making it
easier to recover the original variable. Our recovering technique also provides
a simple and generic framework to solve general ill-posed problems in a
computationally stable way.
