Abstract: We derive a mathematical mechanism of infrared
catastrophe in a general framework. We define the Carleman operator
from the operator-theoretical pull-through formula on ground states.
Characterizing infrared catastrophe by this operator, we study the
mechanism of infrared catastrophe through the domain of the
Carleman operator. As applications of our results, we consider the
generalized spin-boson model and generalized Lee-Pines model. For
the latter model we prove that the infrared singularity condition
causes not only the infrared catastrophe but also the infinite
uncertainty of the position of the nonrelativistic particle
dressing itself in the cloud of soft bosons indirectly.