  Using the pseudo-invariant operator method, we investigate the model of a
particle with a time-dependent mass in a complex time-dependent symmetric
potential well $V\left( x,t\right) =if\left(t\right) \left\vert x\right\vert$.
The problem is exactly solvable and the analytic expressions of the
Schr\"{o}dinger wavefunctions are given in terms of the Airy function. Indeed,
with an appropriate choice of the time-dependent metric operators and the
unitary transformations, for each region, the two corresponding
pseudo-Hermitian invariants transform into a well-known time-independent
Hermitian invariant which is the Hamiltonian of a particle confined in a
symmetric linear potential well. The eigenfunctions of the last invariant are
the Airy functions. Then, the phases obtained are real for both regions and the
general solution to the problem is deduced.
