3 problems

Let $q_1$ and $q_2$ be two distinct odd primes, and let $V$ be a nonconstant real-valued $q_1\be Z\binom{}{}\oplus q_2\be Z$-periodic potential. For $[?]$

Conjugate-operator conjecture. There exists such a conjugate operator $\mathbb{A}(n)$ for which the Mourre estimate for $\Delta$ holds with respect to $\mathbb{A}(n)$ for every…

Let $d\geq 2$, let $q\neq 0$ be sign-definite, and let $\omega^{\ast}$ denote the periodic configuration associated with the extremal spectral values $E_{\min}$ and $E_{\max}$. A per…