# 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][4]

In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems.[5][6] 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.[6][7]

The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984.[8][9]

## Context

Background definitions for the "Coulomb energies" problems (${\displaystyle N}$ non-relativistic particles (electrons) in ${\displaystyle \mathbb {R} ^{3}}$ with spin ${\displaystyle 1/2}$ and an infinitely heavy nucleus with charge ${\displaystyle Z}$ and Coulombic mutual interaction):

• ${\displaystyle {\mathcal {H}}_{f}^{(N)}}$ is the space of functions on ${\displaystyle L^{2}(\mathbb {R} ^{3N};\mathbb {C} ^{2N})}$ which are asymmetrical under exchange of the ${\displaystyle N}$ spin and space coordinates.[1] Equivalently, the subspace of ${\displaystyle (L^{2}(\mathbb {R} ^{3})\otimes \mathbb {C} ^{2})^{\otimes N}}$ which is asymmetrical under exchange of the ${\displaystyle N}$ factors.
• The Hamiltonian is ${\displaystyle H(N,Z):=\sum _{i=1}^{N}(-\Delta _{i}-{\frac {Z}{|x_{i}|}})+\sum _{i. Here ${\displaystyle x_{i}\in \mathbb {R} ^{3}}$ is the coordinate of the ${\displaystyle i}$-th particle, ${\displaystyle \Delta _{i}}$ is the Laplacian with respect to the coordinate ${\displaystyle x_{i}}$. Even if the Hamiltonian does not explicitly depend on the state of the spin sector, the presence of spin has an effect due to the asymmetry condition on the total wave-function.
• We define ${\displaystyle E(N,Z):=\min _{{\mathcal {H}}_{f}}H(N,Z)}$, that is, the ground state energy of the ${\displaystyle (N,Z)}$ system.
• We define ${\displaystyle N_{0}(Z)}$ to be the smallest value of ${\displaystyle N}$ such that ${\displaystyle E(N+j,Z)=E(N,Z)}$ for all positive integers ${\displaystyle j}$; it is known that such a number always exists and is always between ${\displaystyle Z}$ and ${\displaystyle 2Z}$, inclusive.[1]

## The 1984 list

Simon listed the following problems in 1984:[8]

No. Short name Statement Status Year solved
1st (a) Almost always global existence for Newtonian gravitating particles (a) Prove that the set of initial conditions for which Newton's equations fail to have global solutions has measure zero.. Open as of 1984.[8][needs update] In 1977, Saari showed that this is true for 4-body problems.[10] ?
(b) Existence of non-collisional singularities in the Newtonian N-body problem Show that there are non-collisional singularities in the Newtonian N-body problem for some N and suitable masses. In 1988, Xia gave an example of a 5-body configuration which undergoes a non-collisional singularity.[11][12]

In 1991, Gerver showed that 3n-body problems in the plane for some sufficiently large value of n also undergo non-collisional singularities.[13]

1989
2nd (a) Ergodicity of gases with soft cores Find repulsive smooth potentials for which the dynamics of N particles in a box (with, e.g., smooth wall potentials) is ergodic. Open as of 1984.[needs update]

Sinai once proved that the hard sphere gas is ergodic, but no complete proof has appeared except for the case of two particles, and a sketch for three, four, and five particles.[8]

?
(b) Approach to equilibrium Use the scenario above to justify that large systems with forces that are attractive at suitable distances approach equilibrium, or find an alternate scenario that does not rely on strict ergodicity in finite volume. Open as of 1984.[needs update] ?
(c) Asymptotic abelianness for the quantum Heisenberg dynamics Prove or disprove that the multidimensional quantum Heisenberg model is asymptotically abelian. Open as of 1984.[needs update] ?
3rd Turbulence and all that Develop a comprehensive theory of long-time behavior of dynamical systems, including a theory of the onset of and of fully developed turbulence. Open as of 1984.[needs update] ?
4th (a) Fourier's heat law Find a mechanical model in which a system of size ${\displaystyle L}$ with temperature difference ${\displaystyle \Delta T}$ between its ends has a rate of heat temperature that goes as ${\displaystyle L^{-1}}$ in the limit ${\displaystyle L\to \infty }$. Open as of 1984.[needs update] ?
(b) Kubo's formula Justify Kubo's formula in a quantum model or find an alternate theory of conductivity. Open as of 1984.[needs update] ?
5th (a) Exponential decay of ${\displaystyle v=2}$ classical Heisenberg correlations Consider the two-dimensional classical Heisenberg model. Prove that for any beta, correlations decay exponentially as distance approaches infinity. Open as of 1984.[needs update] ?
(b) Pure phases and low temperatures for the ${\displaystyle v\geq 3}$ classical Heisenberg model Prove that, in the ${\displaystyle D=3}$ model at large beta and at dimension ${\displaystyle v\geq 3}$, the equilibrium states form a single orbit under ${\displaystyle SO(3)}$: the sphere.
(c) GKS for classical Heisenberg models Let ${\displaystyle f}$ and ${\displaystyle g}$ be finite products of the form ${\displaystyle (\sigma _{\alpha }\cdot \sigma _{\gamma })}$ in the ${\displaystyle D=3}$ model. Is it true that ${\displaystyle _{\Lambda ,\beta }\geq _{\Lambda ,\beta }_{\Lambda ,\beta }}$ ?[clarification needed]
(d) Phase transitions in the quantum Heisenberg model Prove that for ${\displaystyle v\geq 3}$ and large beta, the quantum Heisenberg model has long range order.
6th Explanation of ferromagnetism Verify the Heisenberg picture of the origin of ferromagnetism (or an alternative) in a suitable model of a realistic quantum system. Open as of 1984.[needs update] ?
7th Existence of continuum phase transitions Show that for suitable choices of pair potential and density, the free energy is non-${\displaystyle C^{1}}$ at some beta. Open as of 1984.[needs update] ?
8th (a) Formulation of the renormalization group Develop mathematically precise renormalization transformations for ${\displaystyle v}$-dimensional Ising-type systems. Open as of 1984.[needs update] ?
(b) Proof of universality Show that critical exponents for Ising-type systems with nearest neighbor coupling but different bond strengths in the three directions are independent of ratios of bond strengths.
9th (a) Asymptotic completeness for short-range N-body quantum systems Prove that ${\displaystyle \oplus ~{\text{Ran}}~\Omega _{a}^{+}=L^{2}(X)}$.[clarification needed] Open as of 1984.[8][needs update] ?
(b) Asymptotic completeness for Coulomb potentials Suppose ${\displaystyle v=3,V_{ij}(x)=e_{ij}|x|^{-1}}$. Prove that ${\displaystyle \oplus ~{\text{Ran}}~\Omega _{a}^{D,+}=L^{2}(X)}$.[clarification needed]
10th (a) Monotonicity of ionization energy (a) Prove that ${\displaystyle (\Delta E)(N-1,Z)\geq (\Delta E)(N,Z)}$.[clarification needed] Open as of 1984.[needs update] ?
(b) The Scott correction Prove that ${\displaystyle \lim _{Z\to \infty }(E(Z,Z)-e_{TF}Z^{7/3})/Z^{2}}$ exists and is the constant found by Scott.[clarification needed]
(c) Asymptotic ionization Find the leading asymptotics of ${\displaystyle (\Delta E)(Z,Z)}$.[clarification needed]
(d) Asymptotics of maximal ionized charge Prove that ${\displaystyle \lim _{Z\to \infty }N(Z)/Z=1}$.[clarification needed]
(e) Rate of collapse of Bose matter Find suitable ${\displaystyle C_{1},C_{2},\alpha }$ such that ${\displaystyle -C_{1}N^{\alpha }\leq {\tilde {E}}_{B}(N,N;1)\leq C_{2}N^{\alpha }}$.[clarification needed]
11th Existence of crystals Prove a suitable version of the existence of crystals (e.g. there is a choice of minimizing configurations that converge to some infinite lattice configuration). Open as of 1984.[needs update] ?
12th (a) Existence of extended states in the Anderson model Prove that in ${\displaystyle v\geq 3}$ and for small ${\displaystyle \lambda }$ that there is a region of absolutely continuous spectrum of the Anderson model, and determine whether this is false for ${\displaystyle v=2}$.[clarification needed] Open as of 1984.[needs update] ?
(b) Diffusive bound on "transport" in random potentials Prove that ${\displaystyle {\text{Exp}}(\delta _{0},(e^{itH}{\vec {N}}e^{-itH})^{2}\delta _{0})\leq c(1+|t|)}$ for the Anderson model, and more general random potentials.[clarification needed]
(c) Smoothness of ${\displaystyle k(E)}$ through the mobility edge in the Anderson model Is ${\displaystyle k(E)}$, the integrated density of states[clarification needed], a ${\displaystyle C^{\infty }}$ function in the Anderson model at all couplings?
(d) Analysis of the almost Mathieu equation Verify the following for the almost Mathieu equation:
• If ${\displaystyle \alpha }$ is a Liouville number and ${\displaystyle \lambda \neq 0}$, then the spectrum is purely singular continuous for almost all ${\displaystyle \theta }$.
• If ${\displaystyle \alpha }$ is a Roth number and ${\displaystyle |\lambda |<2}$, then the spectrum is purely absolutely continuous for almost all ${\displaystyle \theta }$.
• If ${\displaystyle \alpha }$ is a Roth number and ${\displaystyle |\lambda |>2}$, then the spectrum is purely dense pure point.
• If ${\displaystyle \alpha }$ is a Roth number and ${\displaystyle |\lambda |=2}$, then ${\displaystyle \sigma (h)}$has Lebesgue measure zero and the spectrum is purely singular continuous.[clarification needed]
(e) Point spectrum in a continuous almost periodic model Show that ${\displaystyle -{\frac {d^{2}}{dx^{2}}}+\lambda \cos(2\pi x)+\mu \cos(2\pi \alpha x+\theta )}$ has some point spectrum for suitable ${\displaystyle \alpha ,\lambda ,\mu }$ and almost all ${\displaystyle \theta }$.
13th Critical exponent for self-avoiding walks Let ${\displaystyle D(n)}$ be the mean displacement of a random self-avoiding walk of length ${\displaystyle n}$. Show that ${\displaystyle v:=\lim _{n\to \infty }n^{-1}\ln D(n)}$ is ${\displaystyle {\frac {1}{2}}}$ for dimension at least four and is greater otherwise. Open as of 1984.[needs update] ?
14th (a) Construct QCD Give a precise mathematical construction of quantum chromodynamics. Open as of 1984.[needs update] ?
(b) Renormalizable QFT Construct a nontrivial quantum field theory that is renormalizable but not superrenormalizable.
(c) Inconsistency of QED Prove that QED is not a consistent theory.
(d) Inconsistency of ${\displaystyle \varphi _{4}^{4}}$ Prove that a nontrivial ${\displaystyle \varphi _{4}^{4}}$ theory does not exist.
15th Cosmic censorship Formulate and then prove or disprove a suitable version of cosmic censorship. Open as of 1984.[needs update] ?

In 2000, Simon claimed that five[which?] of the problems he listed had been solved.[1]

## The 2000 list

The Simon problems as listed in 2000 (with original categorizations) are:[1][14]

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 ${\displaystyle v\geq 3}$ and suitable values of ${\displaystyle b-a}$ in some energy range. ? ?
2nd Localization in 2 dimensions Prove that the spectrum of the Anderson model for ${\displaystyle v=2}$ is dense pure point. ? ?
3rd Quantum diffusion Prove that, for ${\displaystyle v\geq 3}$ and values of ${\displaystyle |b-a|}$ where there is absolutely continuous spectrum, that ${\displaystyle \sum _{n\in \mathbb {Z} ^{\nu }}n^{2}|e^{itH}(n,0)|^{2}}$ grows like ${\displaystyle ct}$ as ${\displaystyle t\to \infty }$. ? ?
4th Ten Martini problem Prove that the spectrum of ${\displaystyle h_{\alpha ,\lambda ,\theta }}$ is a Cantor set (that is, nowhere dense) for all ${\displaystyle \lambda \neq 0}$ and all irrational ${\displaystyle \alpha }$. Solved by Puig (2003).[14][15] 2003
5th Prove that the spectrum of ${\displaystyle h_{\alpha ,\lambda ,\theta }}$ has measure zero for ${\displaystyle \lambda =2}$ and all irrational ${\displaystyle \alpha }$. Solved by Avila and Krikorian (2003).[14][16] 2003
6th Prove that the spectrum of ${\displaystyle h_{\alpha ,\lambda ,\theta }}$ is absolutely continuous for ${\displaystyle \lambda =2}$ and all irrational ${\displaystyle \alpha }$. ? ?
7th Do there exist potentials ${\displaystyle V(x)}$ on ${\displaystyle [0,\infty )}$ such that ${\displaystyle |V(x)|\leq C|x|^{{\frac {1}{2}}+\varepsilon }}$ for some ${\displaystyle \varepsilon }$ and such that ${\displaystyle -{\frac {d^{2}}{dx^{2}}}+V}$ has some singular continuous spectrum? Essentially solved by Denisov (2003) with only ${\displaystyle L^{2}}$ decay.

Solved entirely by Kiselev (2005).[14][17][18]

2003, 2005
8th Suppose that ${\displaystyle V(x)}$ is a function on ${\displaystyle \mathbb {R} ^{\nu }}$ such that ${\displaystyle \int |x|^{-\nu +1}|V(x)|^{2}d^{\nu }x<\infty }$, where ${\displaystyle \nu \geq 2}$. Prove that ${\displaystyle -\Delta +V}$ has absolutely continuous spectrum of infinite multiplicity on ${\displaystyle [0,\infty )}$. ? ?
Coulomb energies
9th Prove that ${\displaystyle N_{0}(Z)-Z}$ is bounded for ${\displaystyle Z\to \infty }$. ? ?
10th What are the asymptotics of${\displaystyle (\delta E)(Z):=E(Z,Z-1)-E(Z,Z)}$ for ${\displaystyle Z\to \infty }$? ? ?
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 ${\displaystyle k(E)}$ is continuous in the energy. | k(E1 + ΔE) - k(E1) | < ε ?
15th Lieb-Thirring conjecture Prove the Lieb-Thirring conjecture on the constants ${\displaystyle L_{\gamma ,\nu }}$ where ${\displaystyle \nu =1,{\frac {1}{2}}<\gamma <{\frac {3}{2}}}$. ? ?