6 problems

Let $H_0=-\Delta+q(x)$ be a Schrödinger operator with periodic potential $q$, let $F_{\,\lambda}(q)$ be its Fermi variety, and let the Condition mean that, for every $\lambda$ in t…

Approximation property. Any smooth potential can be approximated by algebraic-geometrical ones with the same periods.

Let $q>3$, let $V$ be a nonzero $q\mathbb{Z}$-periodic potential of the form … and let $\Delta$ denote the discrete Laplacian. Floquet-isospectral potential conjecture. There exist…

Let $d\geq1$ and let $\gamma$ satisfy the constraint defining the admissible Lieb–Thirring regime. Let $L_{\gamma,d}$ be the optimal Lieb–Thirring constant, let…

Let $d\geq 2$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be a periodic potential such that the Schrödinger operator $H_V$ is self-adjoint. Bethe–Sommerfeld conjecture. The spectrum…

Let $m$ denote the number of Fourier modes in the periodic potentials, let $n$ be the ambient dimension, and let $V_1,V_2$ be smooth periodic functions. The rigidity result referre…