Simon problems

In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist.^[1]^[2] Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum operators.^[3] Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential.^[1]

In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems.^[4]^[5] Among these was the problem of proving that the set of energy levels of one particular abstract quantum system was in fact the Cantor set, a challenge known as the "Ten Martini Problem" after the reward that Mark Kac offered for solving it.^[5]^[6]

The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984.^[7]^[8]

Context

Background definitions for the "Coulomb energies" problems:

${\mathcal {H}}_{f}^{(N)}$ is the space of functions on $L^{2}({\mathcal {R}}^{2N};{\mathcal {C}}^{3N})$ which are antisymmetric in spin and space.^[1]^{[clarification needed]}
The Hamiltonian is ${\mathcal {H}}(N,Z):=\sum _{i=1}^{N}(-\Delta _{i}-{\frac {Z}{x_{i}}})+\sum _{i<j}{\frac {1}{|x_{i}-x_{j}|}}$ .^[1]^{[clarification needed]}
We have $E(N,Z):=\min _{{\mathcal {H}}_{f}}H(N,Z)$ .^[1]^{[clarification needed]}
We define $N_{0}(Z)$ to be the smallest value of $E(N+j,Z)=E(N,Z)$ for all positive integers $j$ ; it is known that such a number always exists and is always between $Z$ and $2Z$ , inclusive.^[1]

The 2000 list

The Simon problems as listed in 2000 (with original categorizations) are:^[1]^[9]

No.	Short name	Statement	Status	Year solved
Quantum transport and anomalous spectral behavior
1st	Extended states	Prove that the Anderson model has purely absolutely continuous spectrum for $v\geq 3$ and suitable values of $b-a$ in some energy range.	?	?
2nd	Localization in 2 dimensions	Prove that the spectrum of the Anderson model for $v=2$ is dense pure point.	?	?
3rd	Quantum diffusion	Prove that, for $v\geq 3$ and values of $\|b-a\|$ where there is absolutely continuous spectrum, that $\sum _{n\in \mathbb {Z} ^{\nu }}n^{2}\|e^{itH}(n,0)\|^{2}$ grows like $ct$ as $t\to \infty$ .	Solved by Puig (2003).^[9]^[10]	2003
4th	Ten Martini problem	Prove that the spectrum of $h_{\alpha ,\lambda ,\theta }$ is a Cantor set (that is, nowhere dense) for all $\lambda \neq 0$ and all irrational $\alpha$ .	?	?
5th		Prove that the spectrum of $h_{\alpha ,\lambda ,\theta }$ has measure zero for $\lambda =2$ and all irrational $\alpha$ .	Solved by Avila and Krikorian (2003).^[9]^[11]	2003
6th		Prove that the spectrum of $h_{\alpha ,\lambda ,\theta }$ is absolutely continuous for $\lambda =2$ and all irrational $\alpha$ .	?	?
7th		Do there exist potentials $V(x)$ on $[0,\infty )$ such that $\|V(x)\|\leq C\|x\|^{{\frac {1}{2}}+\varepsilon }$ for some $\varepsilon$ and such that $-{\frac {d^{2}}{dx^{2}}}+V$ has some singular continuous spectrum?	Essentially solved by Denisov (2003) with only $L^{2}$ decay. Solved entirely by Kiselev (2005).^[9]^[12]^[13]	2003, 2005
8th		Suppose that $V(x)$ is a function on $\mathbb {R} ^{\nu }$ such that $\int \|x\|^{-\nu +1}\|V(x)\|^{2}d^{\nu }x<\infty$ , where $\nu \geq 2$ . Prove that $-\Delta +V$ has absolutely continuous spectrum of infinite multiplicity on $[0,\infty )$ .	?	?
Coulomb energies
9th		Prove that $N_{0}(Z)-Z$ is bounded for $Z\to \infty$ .	?	?
10th		What are the asymptotics of $(\delta E)(Z):=E(Z,Z-1)-E(Z,Z)$ for $Z\to \infty$ ?	?	?
11th		Make mathematical sense of the nuclear shell model.	?	?
12th		Is there a mathematical sense in which one can justify current techniques for determining molecular configurations from first principles?	?	?
13th		Prove that, as the number of nuclei approaches infinity, the ground state of some neutral system of molecules and electrons approaches a periodic limit (i.e. that crystals exist based on quantum principles).	?	?
Other problems
14th		Prove that the integrated density of states $k(E)$ is continuous in the energy.	?	?
15th	Lieb-Thirring conjecture	Prove the Lieb-Thirring conjecture on the constants $L_{\gamma ,\nu }$ where $\nu =1,{\frac {1}{2}}<\gamma <{\frac {3}{2}}$ .	?	?