Simon problems
In mathematics, the Simon 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. Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential.
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.
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 :- is the space of functions on which are asymmetrical under exchange of the spin and space coordinates. Equivalently, the subspace of which is asymmetrical under the exchange of the factors.
- The Hamiltonian is. Here is the coordinate of the -th particle, is the Laplacian with respect to the coordinate. 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, that is, the ground state energy of the system.
- We define to be the smallest value of such that for all positive integers ; it is known that such a number always exists and is always between and, inclusive.
The 1984 list
Simon listed the following problems in 1984:| No. | Short name | Statement | Status | Year solved |
| 1st | Almost always global existence for Newtonian gravitating particles | Prove that the set of initial conditions for which Newton's equations fail to have global solutions has measure zero. | Open as of 1984. In 1977, Saari showed that this is true for 4-body problems. | ? |
| 1st | 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. In 1991, Gerver showed that 3n-body problems in the plane for some sufficiently large value of n also undergo non-collisional singularities. | 1989 |
| 2nd | Ergodicity of gases with soft cores | Find repulsive smooth potentials for which the dynamics of N particles in a box is ergodic. | Open as of 1984. 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. | ? |
| 2nd | 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. | ? |
| 2nd | Asymptotic abelianness for the quantum Heisenberg dynamics | Prove or disprove that the multidimensional quantum Heisenberg model is asymptotically abelian. | Open as of 1984. | ? |
| 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. | ? |
| 4th | Fourier's heat law | Find a mechanical model in which a system of size with temperature difference between its ends has a rate of heat temperature that goes as in the limit. | Open as of 1984. | ? |
| 4th | Kubo's formula | Justify Kubo's formula in a quantum model or find an alternate theory of conductivity. | Open as of 1984. | ? |
| 5th | Exponential decay of 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. | ? |
| 5th | Pure phases and low temperatures for the classical Heisenberg model | Prove that, in the model at large beta and at dimension, the equilibrium states form a single orbit under : the sphere. | Open as of 1984. | ? |
| 5th | GKS for classical Heisenberg models | Let and be finite products of the form in the model. Is it true that ? | Open as of 1984. | ? |
| 5th | Phase transitions in the quantum Heisenberg model | Prove that for and large beta, the quantum Heisenberg model has long range order. | Open as of 1984. | ? |
| 6th | Explanation of ferromagnetism | Verify the Heisenberg picture of the origin of ferromagnetism in a suitable model of a realistic quantum system. | Open as of 1984. | ? |
| 7th | Existence of continuum phase transitions | Show that for suitable choices of pair potential and density, the free energy is non- at some beta. | Open as of 1984. | ? |
| 8th | Formulation of the renormalization group | Develop mathematically precise renormalization transformations for -dimensional Ising-type systems. | Open as of 1984. | ? |
| 8th | 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. | Open as of 1984. | ? |
| 9th | Asymptotic completeness for short-range N-body quantum systems | Prove that. | Open as of 1984. | ? |
| 9th | Asymptotic completeness for Coulomb potentials | Suppose. Prove that. | Open as of 1984. | ? |
| 10th | Monotonicity of ionization energy | Prove that. | Open as of 1984. | ? |
| 10th | The Scott correction | Prove that exists and is the constant found by Scott. | Open as of 1984. | ? |
| 10th | Asymptotic ionization | Find the leading asymptotics of. | Open as of 1984. | ? |
| 10th | Asymptotics of maximal ionized charge | Prove that. | Open as of 1984. | ? |
| 10th | Rate of collapse of Bose matter | Find suitable such that. | Open as of 1984. | ? |
| 11th | Existence of crystals | Prove a suitable version of the existence of crystals. | Open as of 1984. | ? |
| 12th | Existence of extended states in the Anderson model | Prove that in and for small that there is a region of absolutely continuous spectrum of the Anderson model, and determine whether this is false for. | Open as of 1984. | ? |
| 12th | Diffusive bound on "transport" in random potentials | Prove that for the Anderson model, and more general random potentials. | Open as of 1984. | ? |
| 12th | Smoothness of through the mobility edge in the Anderson model | Is, the integrated density of states, a function in the Anderson model at all couplings? | Open as of 1984. | ? |
| 12th | Analysis of the almost Mathieu equation | Verify the following for the almost Mathieu equation:
| Open as of 1984. | ? |
| 12th | Point spectrum in a continuous almost periodic model | Show that has some point spectrum for suitable and almost all. | Open as of 1984. | ? |
| 13th | Critical exponent for self-avoiding walks | Let be the mean displacement of a random self-avoiding walk of length. Show that is for dimension at least four and is greater otherwise. | Open as of 1984. | ? |
| 14th | Construct QCD | Give a precise mathematical construction of quantum chromodynamics. | Open as of 1984. | ? |
| 14th | Renormalizable QFT | Construct a nontrivial quantum field theory that is renormalizable but not superrenormalizable. | Open as of 1984. | ? |
| 14th | Inconsistency of QED | Prove that QED is not a consistent theory. | Open as of 1984. | ? |
| 14th | Inconsistency of | Prove that a nontrivial theory does not exist. | Open as of 1984. | ? |
| 15th | Cosmic censorship | Formulate and then prove or disprove a suitable version of cosmic censorship. | Open as of 1984. | ? |