Essential self-adjointness of non-semibounded Schrödinger operators on infinite graphs

Abstract

We work in the setting of infinite, not necessarily locally finite, weighted graphs. We give a sufficient condition for the essential self-adjointness of (discrete) Schrödinger operators LV that are not necessarily lower semi-bounded. As a corollary of the main result, we show that LV is essentially self-adjoint if the potential V satisfies V(x)≥−b1−b2ρ(o,x), for all vertices x∈X, where o is a fixed vertex, b1 and b2 are non-negative constants, and ρ is an intrinsic metric satisfying the condition (B*): restriction of the weighted vertex degree to every ball corresponding to ρ is bounded (not necessarily uniformly bounded). Furthermore, we show that LV is essentially self-adjoint if V(x)≥−b1−b2[ρ(o,x)]2 for all x∈X, where o is a fixed vertex, b1 and b2 are non-negative constants, and ρ is an intrinsic metric having a finite jump size and satisfying the condition (B*).