The finite-combination Mourre conjecture between consecutive thresholds

Let $D$ be the discrete Molchanov–Vainberg Laplacian in dimension $2$, let $\kappa$ be even, and let $\mathcal{E}_{i_1},\mathcal{E}_{i_2}\in\boldsymbol{\Theta}_{\kappa}(D)$ be consecutive thresholds, meaning that no threshold lies strictly between them. Let $A_{j\kappa}$ be the conjugate operators defined in the paper. Consecutive-threshold Mourre conjecture. There is a finite linear combination

$$\mathbb{A}=\sum_{j=1}^{N}\rho_{j\kappa}A_{j\kappa}$$

such that the Mourre estimate holds with $\mathbb{A}$ for every $E\in(\mathcal{E}_{i_1},\mathcal{E}_{i_2})$. Consequently, $\sigma_p(D+V)$ is locally finite on this interval and the singular continuous spectrum of $D+V$ is empty there. The conjecture is stated as an overall dimension-two spectral picture and is not proved in the supplied text.