Simon problems

Not to be confused with Simon's problem in computational complexity.

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. Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum operators.

In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems. 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. Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential.

The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984.

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.
• The Hamiltonian is ${\mathcal {H}}(N,Z):=\sum _{i=1}^{N}(-\Delta _{i}-{\frac {Z}{x_{i}}})+\sum _{i<j}{\frac {1}{|x_{i}-x_{j}|}}$.
• We have $E(N,Z):=\min _{{\mathcal {H}}_{f}}H(N,Z)$.
• 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.

The 2000 list

The Simon problems as listed in 2000 (with categorization) are:

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 $t\to \infty$.	?	?
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 $\alpha$.	?	?
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?	?	?
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 $Z\to \infty$?	?	?

See also

• Almost Mathieu operator
• Lieb–Thirring inequality

External links

• "Simon's Problems". MathWorld. Retrieved 2018-06-13.