2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99755We mathematically study the infrared catastrophe for the Hamiltonian of Nelson's model when it has the external potential in a general class. For the model, we prove the pull-through formula on ground states in operator theory first. Based on this formula, we show both non-existence of any ground state and divergence of the total number of soft bosons.22 pages. This manuscript is the final version and it appears in Publ. RIMSMathematical PhysicsFunctional Analysis81T190, 81V10text