Abstract In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schrödinger operators. Schrödinger operator S is classified as subcritical/critical in terms of the existence/nonexistence of a positive Green function for the associated elliptic equation Su=f S u = f. Such a property strongly affects to the large-time behavior of solutions to the parabolic equation ₜv+Sv=0 ∂ t v + S v = 0. In this paper, we propose a remarkable quantity in terms of the structure of Hilbert lattices, which keeps some important properties including the notion of criticality theory. As an application, we study the large-time behavior of solutions to the hyperbolic equation ₜ²w+Sw=0 ∂ t 2 w + S w = 0.
Motohiro Sobajima (Thu,) studied this question.