Neutron falling off a cliff edge

Consider a particle, say a neutron, of mass $m$ moving from some $x < 0$ with some speed $ v_0 $ in the positive $x$ direction along the ground, see Figure 1. At $ x = 0 $ there is a drop of height $ h $ and the particle’s motion for $ x > 0 $ is influenced by gravity, see https://www.youtube.com/watch?v=svOMRjp4kRw

I wish to determine the solutions to the time independent Schrodinger equation
begin{equation*}
left( -frac{hbar^2}{2 m} frac{ partial^2}{partial x^2} -frac{hbar^2}{2 m} frac{ partial^2}{partial z^2} + V ( x, z ) right) psi left( x, z right) = E psi left( x, z right).
end{equation*}
I thought of taking the potential $V ( x, z ) $ to be of the form
begin{equation*}
V ( x, z ) = C – m g z Theta ( x )
end{equation*}
so that the total energy is $ E = C + frac{1}{2} m v_0^2. $ . I assumed that it would be reasonable to then consider the limit as $ C to infty $ to take into account that the ground forms as impassable barrier. For finite $C$ I think the potential looks like that depicted in the following figure

For $x le 0$ Schodinger’s equation looks like
begin{equation*}
left( -frac{hbar^2}{2 m} frac{ partial^2}{partial x^2} -frac{hbar^2}{2 m} frac{ partial^2}{partial z^2} + C right) psi left( x, z right) = E psi left( x, z right)
end{equation*}
and I assume that solutions can be written in the form
begin{equation*}
psi left( x, z right) = left( e^{ i k x } + alpha e^{ – i k x} right) u_1 ( z )
end{equation*}
with $hbar k = m v_0$, that is, a forward travelling wave with the possibility of a reflected wave. For $ x > 0$ Schrodinger’s equation becomes
begin{equation*}
left( -frac{hbar^2}{2 m} frac{ partial^2}{partial x^2} -frac{hbar^2}{2 m} frac{ partial^2}{partial z^2} + C – m g z right) psi left( x, z right) = E psi left( x, z right)
end{equation*}
and we look for solutions of the form a plane wave travelling in the $+ve ; x$ modulated by $ u_2 ( z ) $ that will be a linear combination of the Airy functions $A_i ( z ), B_i ( z )$
begin{equation*}
psi ( x, z ) = u_2 ( z ) e^{i k x },
end{equation*}
For $x > 0$ taking the limit $C to infty$ doesn’t seem to introduce any problems, simply implying the boundary condition $u_2 ( – h ) = 0.$ However there seems to be problems for $x < 0$ since it seems to imply $ u_1 ( z ) equiv 0 $.newline
How does one solve this problem?