Remarks on criticality theory for Schrödinger operators and its application to wave equations with potentials
Motohiro Sobajima
TL;DR
This work reframes criticality theory for nonnegative Schrödinger operators through a Hilbert-lattice perspective, introducing the interval $I_S$ of exponents $\alpha$ for which the range $R(S^{\alpha})$ intersects the positive cone. It proves an equivalence between the nontrivial positive intersection $R(S^{\alpha})\cap [L^2(\Omega)]_{+}$ and the density of $C_0^{\infty}(\Omega)$ in $R(S^{\alpha})$, with $\alpha=\tfrac{1}{2}$ characterizing subcriticality, and develops a general theory for $R(A^{\alpha})$ in Hilbert spaces, including density and interpolation properties and a Hadamard-type relation linking heat and wave evolutions. These results are then applied to wave equations with potentials, yielding boundedness and decay estimates for solutions, and a detailed analysis of inverse-square potentials highlighting threshold behavior tied to Hardy inequalities and dimension. Overall, the paper provides a functional-analytic framework to study long-time dynamics of wave-type PDEs with potentials and extends criticality concepts beyond classical heat-kernel criteria.
Abstract
In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schrödinger operators. Schrödinger operator $S=-Δ+V$ is classified as subcritical/critical in terms of the existence/nonexistence of a positive Green function for the associated elliptic equation $Su=f$. Such a property strongly affects to the large-time behavior of solutions to the parabolic equation $\partial_tv+Sv=0$. 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 $\partial_t^2w+Sw=0$.