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We propose a probabilistic representation of the ground states of massive andmassless Schr\"{o}dinger operators with a potential well in which the behaviourinside the well is described in terms of the moment generating function of thefirst exit time from the well, and the outside behaviour in terms of theLaplace transform of the first entrance time into the well. This allows ananalysis of their behaviour at short to mid-range from the origin. In a firstpart we derive precise estimates on these two functionals for stable andrelativistic stable processes. Next, by combining scaling properties and heatkernel estimates, we derive explicit local rates of the ground states of thegiven family of non-local Schr\"{o}dinger operators both inside and outside thewell. We also show how this approach extends to fully supported decayingpotentials. By an analysis close-by to the edge of the potential well, wefurthermore show that the ground state changes regularity, which dependsqualitatively on the fractional power of the non-local operator.

Bulk behaviour of ground states for relativistic Schr\"odinger operators with compactly supported potentials