Small perturbations leading to large asymmetries in the probability density of the ground state

It is shown that in one-particle Schr\"{o}dinger quantum mechanics a small perturbation of a one-dimensional potential can produce a large change in the ground state, the effect becoming more pronounced with growing typical length of the potential, contrary to what standard stationary perturbation theory could make one believe.The possible consequences of this result for solid state physics and other fields are also discussed, and examples of such non-standard perturbations are furnished.