Well, generally it's not true. You need to start the particle in a state where it's true and have conditions (at infinity) on the potential V that can keep the particle from escaping to [$]\infty[$].
Assuming all that (i.e., there are only bound state solutions), then for each such solution:
[$]|\psi(t,x)|^2[$] is the probability density for finding the particle at [$]x[$],
[$]\Rightarrow \int_{-\infty}^{\infty} |\psi(t,x)|^2 \, dx = 1[$]
[$]\Rightarrow \psi \rightarrow 0[$] as [$]x \rightarrow \pm\infty[$]
But, if V=0, then [$]\psi(t,x) = e^{\pm i \sqrt{c} x - i c t} [$] for say some positive c and you can see the problem. That would be a scattering state (aka a travelling wave), ignoring the problem that it's not normalizable. Many potentials V will allow similar behavior for [$]\psi[$] far away from the barrier. For example, if V is finite and "too shallow", then regardless of how you prepare the particle at t=0 (the initial condition), it might escape the region and become a scattering state that heads off to [$]\pm \infty[$].
Yes, demanding that [$]\psi[$] tapers off at the infinities is a bit rough-and-ready (I got it from Numerical Recipes (NR)) and it will not work as you say (also not numerically, more later). The mathematical details are missing. If you reason in probabilities then the solution will have compact support but is a bit of hand-waving.
One point is that these PDEs are travelling waves in possibly infinite domains. I looked up my QM notes from 1974 (lol) and found my solution for the time-independent Schroedinger equation with a rectangular potential barrier. Insightful.
For most problems we need numerical FDM and Absorbing BC it seems. So we want the travelling waves to pass through the artificial boundary with no reflection (like with non-quantum wave equation). This results in 2 1st order hyperbolic PDEs on the boundaries. Which PDE + BC can be solved using Crank Nicolson as mentioned in NR.
I think that the infinite domain is truncated and the ABC is placed on the truncated boundaries? Would domain transformation not be a better option?
On a time-dependent interval [$](0, L(t))[$] using Landau transform [$] y = x/L(t)[$] to get a PDE with convection term on [$](0,1)[$]. The ABC is a "robust" (see Karpova et alia eq. (16) 1st order PDE. See
http://nanojournal.ifmo.ru/en/wp-conten ... P13-19.pdf
BTW equation (13) on RHS ->[$]\frac{\partial \psi}{\partial t}[$] should be [$]\frac{\partial \psi}{\partial y}[$]?
// These problems are somewhat more challenging than normal PDEs here.
// I'm seeing QM from a new perspective.