26 problems

Arbitrary-parameter GAL relation conjecture. Relations $(2.17)$ and $(2.18)$ are valid whenever either $a+b+f+g$ or one of these modified parameter sums is a non-negative even inte…

Odd-sum GAL spectral-correspondence conjecture. The QES mid-band energy values of the displayed transformed potential are the same as the band-edge eigenvalues of $[a,b,f,g]$.

GAL simultaneous-relation conjecture. Regardless of whether $a+b+f+g$ is odd or even, relations $(2.17)$ and $(2.18)$ are both valid.

Half-integer GAL state-count conjecture. This potential has $k+l+n+3$ QES mid-band states.

Half-integer GAL partner-relation conjecture. Relations $(2.17)$ and $(2.18)$ should hold simultaneously whenever at least one of the four parameters $a,b,f,g$ is a half-integer.

GAL integer-to-half-integer correspondence conjecture. Relations $(2.17)$ and $(2.18)$ should hold simultaneously when $a,b,f,g$ are integers, and the band-edge eigenvalues of the…

GAL partner-potential conjecture. The following relationships hold:

Finite-band-gap conjecture. Every GAL potential whose four parameters $a,b,f,g$ are integers has a finite number of band gaps.

Mother Hamiltonian polynomial conjecture. The Mother Hamiltonian is the polynomial

Localized-perturbation conjecture. If, for some $\lambda$ in the interior of a spectral band, there is an irreducible component $A$ of the Fermi surface with…

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…

Let $a$ be an integer, and consider the periodic potential $[(a-1)a,2,(a-2)(a-1),2]$. A potential is finite-gap if it has only finitely many spectral band gaps. Finite-gap conjectu…

Let $a$ be an integer, and consider the associated Lamé potentials denoted by $[a(a+1),a(a+1),a(a+1),(a-1)a]$ and the Lamé potential $[2a(2a+1),0,0,0]$. Their band edges are the sp…

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 conjugate-symmetric potential con…

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 a system of diffusions in a periodic potential be conditioned on never intersecting. The confinement makes it easy for particles in different boxes to stay apart for a long tim…

Let $H$ be a Schrödinger operator with periodic potentials, and consider its bounded solutions. A Bloch solution is a solution exhibiting Bloch-periodic behavior; a limitation mean…

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 $H_0=-\frac{d^2}{dx^2}+V_0$ be a one-dimensional Schrödinger operator with a general non-zero periodic function $V_0$, and let $V$ be a perturbing potential. The preceding crit…

In the periodic quantum transport setting, let the linear Boltzmann equation describe the limiting particle density in the Boltzmann–Grad and high-frequency scaling limits. Higher-…

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…

Let $n\geq 2$, let $H(p)=\frac{1}{2}|p|^2$ for $p\in\mathbb{R}^n$, and let $V_1,V_2\in C(\mathbb{R}^n)$ be $\mathbb{Z}^n$-periodic potential energy functions. Let $\overline{H}_1$…

Let $\psi(x,t)$ be the periodic traveling-wave solution defined by … with the parameter values used in the numerical experiments of. Stability conjecture. The solution $\psi(x,t)$…

Let $\psi(x,t)$ be the periodic traveling-wave solution described above, with nonlocality parameter $\epsilon$ and convolution kernel $R(x;\epsilon)$. Numerical experiments indicat…