It gives a formula if you provide initial conditions:
v[x_] = Piecewise[{{v0, x >= 0}}, 0];
stationarySchroedingerEquation = (-\[HBar]^2/(2 m)) \[Psi]''[x] +
v[x] \[Psi][x] == E*\[Psi][x];
sol = DSolveValue[{stationarySchroedingerEquation, \[Psi][
0] == \[Psi]0, \[Psi]'[0] == \[Psi]1}, \[Psi], x]
Simplify[stationarySchroedingerEquation /. \[Psi] -> sol]
ReImPlot[
Evaluate[
sol[x] /. {m -> 1,
v0 -> 1, \[Psi]0 -> 2, \[Psi]1 -> -1, \[HBar] -> 1}], {x, -10, 10}]
