Criticality of Generalized Schrödinger Operators and Differentiability of Spectral Functions

Abstract

Let $\mu$ be a positive Radon measure in the Kato class. We consider the spectral bound $C(\lambda) = -\inf \sigma(\mathcal{H}^{\lambda \mu})\ (\lambda \in \mathbb{R}^1)$ of a generalized Schrödinger operator $\mathcal{H}^{\lambda \mu} = -\frac{1}{2} \Delta - \lambda \mu$ on $\mathbb{R}^d$, and show that the spectral bound is differentiable if $d \le 4$ and $\mu$ is Green-tight.