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. Eight of the problems pertain to the 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 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 Mark Kac offered for solving it.
The 2000 list was a refinement of a similar set of problems Simon had posed in 1984.
Context Background definitions for the "Coulomb energies" problems ( non-relativistic particles (electrons) in with spin and an infinitely heavy nucleus with charge and Coulombic mutual interaction):
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 such a number always exists and is always between and , inclusive.
The 1984 list Simon listed the following problems in 1984: No.Short nameStatementStatusYear solved1st(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. In 1977, Donald Saari showed this is true for 4-body problems.?(b) Existence of non-collisional singularities in the Newtonian N-body problemShow 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 3n-body problems in the plane for some sufficiently large value of n also undergo non-collisional singularities.19892nd(a) Ergodicity of gases with soft coresFind 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. Yakov Sinai once proved 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.?(b) Approach to equilibriumUse 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.?(c) Asymptotic abelianness for the quantum Heisenberg dynamicsProve or disprove that the multidimensional quantum Heisenberg model is asymptotically abelian.Open as of 1984.?3rdTurbulence and all thatDevelop 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(a) Fourier's heat lawFind 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.?(b) Kubo's formulaJustify Kubo's formula in a quantum model or find an alternate theory of conductivity.Open as of 1984.?5th(a) Exponential decay of classical Heisenberg correlationsConsider the two-dimensional classical Heisenberg model. Prove that for any beta, correlations decay exponentially as distance approaches infinity.Open as of 1984.?(b) Pure phases and low temperatures for the classical Heisenberg modelProve that, in the model at large beta and at dimension , the equilibrium states form a single orbit under : the sphere.(c) GKS for classical Heisenberg modelsLet and be finite products of the form in the model. Is it true that ?(d) Phase transitions in the quantum Heisenberg modelProve that for and large beta, the quantum Heisenberg model has long range order.6thExplanation of ferromagnetismVerify the Heisenberg picture of the origin of ferromagnetism (or an alternative) in a suitable model of a realistic quantum system.Open as of 1984.?7thExistence of continuum phase transitionsShow that for suitable choices of pair potential and density, the free energy is non- at some beta.Open as of 1984.?8th(a) Formulation of the renormalization groupDevelop mathematically precise renormalization transformations for -dimensional Ising-type systems.Open as of 1984.?(b) Proof of universalityShow 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 systemsProve asymptotic completeness for N-body Schrödinger operators with short-range potentials ().Proved for generic N-body systems in 1987 by I. M. Sigal and A. Soffer; and later by Jan Derezinski and C. Gérard.1987(b) Asymptotic completeness for Coulomb potentialsSuppose . Prove that . Open as of 1984. ?10th(a) Monotonicity of ionization energy(a) Prove that .Open as of 2026.?(b) The Scott correctionProve that the ground state energy of a heavy atom of atomic number Z follows the predicted asymptotic expansion, rigorously justifying the "Scott Correction" to the Thomas-Fermi model.Proved by H. Siedentop and R. Weikard, and later by C. Fefferman and L. Seco.1987(c) Asymptotic ionizationFind the leading asymptotics of . Prove that the maximum number of electrons an atom can bind cannot exceed Z by more than a small amount.Proved by Elliott H. Lieb and Michael Loss.2003(d) Asymptotics of maximal ionized chargeProve that .Proved by Fefferman and Seco.1997(e) Rate of collapse of Bose matterFind suitable such that .Open as of 2026.?11thExistence of crystalsProve 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.?12th(a) Existence of extended states in the Anderson modelProve 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.?(b) Diffusive bound on "transport" in random potentialsProve that for the Anderson model, and more general random potentials.(c) Smoothness of through the mobility edge in the Anderson modelIs , the integrated density of states, a function in the Anderson model at all couplings?(d) Analysis of the almost Mathieu equationVerify the following for the almost Mathieu equation: If is a Liouville number and , then the spectrum is purely singular continuous for almost all . If is a Roth number and , then the spectrum is purely absolutely continuous for almost all . If is a Roth number and , then the spectrum is purely dense pure point. If is a Roth number and , then has Lebesgue measure zero and the spectrum is purely singular continuous.(e) Point spectrum in a continuous almost periodic modelShow that has some point spectrum for suitable and almost all .13thCritical exponent for self-avoiding walksLet 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(a) Construct QCDGive a precise mathematical construction of quantum chromodynamics.Open as of 1984.?(b) Renormalizable QFTConstruct a nontrivial quantum field theory that is renormalizable but not superrenormalizable.(c) Inconsistency of QEDProve that QED is not a consistent theory.(d) Inconsistency of Prove that a nontrivial theory does not exist.15thCosmic censorshipFormulate and then prove or disprove a suitable version of cosmic censorship.Open as of 1984.? In 2000, Simon claimed five of the problems he listed (namely 1B, 9A, 10B, 10C and 10D, 12D) were solved.
The 2000 list The Simon problems as listed in 2000 (with original categorizations), are: No.Short nameStatementStatusYear solvedQuantum transport and anomalous spectral behavior1stExtended statesProve that the Anderson model has purely absolutely continuous spectrum for and suitable values of in some energy range. ??2ndLocalization in 2 dimensionsProve that the spectrum of the Anderson model for is dense pure point.??3rdQuantum diffusionProve that, for and values of where there is absolutely continuous spectrum, that grows like as .??4thTen Martini problemProve that the spectrum of is a Cantor set (that is, nowhere dense) for all and all irrational .Solved by Puig (2003).20035thProve that the spectrum of has measure zero for and all irrational .Solved by Avila and Krikorian (2003).20036thProve that the spectrum of is absolutely continuous for and all irrational .??7thDo there exist potentials on such that for some and such that has some singular continuous spectrum?Essentially solved by Denisov (2003) with only decay. Solved entirely by Kiselev (2005).2003, 20058thSuppose that is a function on such that , where . Prove that has absolutely continuous spectrum of infinite multiplicity on .??Coulomb energies9thProve that is bounded for .??10thWhat are the asymptotics of for ???11thMake mathematical sense of the nuclear shell model.??12thIs there a mathematical sense in which one can justify current techniques for determining molecular configurations from first principles???13thProve 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 problems14thProve that the integrated density of states is continuous in the energy.| k(E1 + ΔE) − k(E1) | < ε?15thLieb-Thirring conjectureProve the Lieb-Thirring conjecture on the constants where .??
See also Almost Mathieu operator Lieb–Thirring inequality
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