  This paper addresses some mathematical problems arising from the infrared
(IR) catastrophe in quantum field theory. IR catastrophe is formulated and
studied in operator theory, characterized by the Carleman operator.
Non-existence of ground state under IR catastrophe is also investigated with
the help of the characterization. The theory presented in this paper is applied
to the Hamiltonian of the model describing a non-relativistic electron coupled
with a quantum field of phonons or polaritons in the light of mathematics as
well as solid state physics.
