Why do we always require the potentials $V(x)$ (both in classical and quantum mechanics) to be bounded from below i.e., require that $V(x)$ should not go to $-\infty$ as $x\rightarrow \pm \infty$?
For example, a potential like $V_1(x)=x^2+x^3$ is often considered "pathological" because it goes to $-\infty$ as $x\rightarrow-\infty$ unlike the potential $V_2(x)=x^2$. It is argued that for $V_1(x)$, the potential has a metastable/local minima, and under small (classical/quantum) perturbation, the particle may roll down to $x\rightarrow -\infty$ in the direction of ever-decreasing potential energy, releasing energy in the form of kinetic energy (if the particle is not electrically charged).
When the particle rolls down to $x\rightarrow -\infty$, the potential energy decreases continuously, and kinetic energy increases from energy conservation. So it doesn't violate the conservation of energy.
Why is this a problem then?