Simon problems

The Simon Problems is a list of fifteen problems from mathematical physics that Barry Simon compiled in 1984 and updated in 2000.

In 1984, Simon listed fifteen problems, most of which each had several sub-problems, so that depending on the count one comes to 35. Some of the problems are extraordinarily difficult fundamental problems and some are deliberately formulated by Simon (following the example of some of Hilbert's problems ) very vaguely and rather outline fields of research.

Problem 1: N-body problem in Newtonian theory of gravity

• Problem 1A: The question is about the existence of global solutions in time. For the set of initial states for which there is no global solution (two or more particles come arbitrarily close in finite time, then potential and velocities diverge), the measure zero in phase space (since it is a subset of the solutions with total angular momentum zero) . Show that this also applies to. ${\displaystyle N=2}$ ${\displaystyle N\geq 3}$

Problems of this kind were dealt with first by Paul Painlevé . He showed that for (and therefore also for fewer particles) all singularities are of the collision type. Zhihong Xia proved in 1992 that there are solutions in which a particle escapes to infinity in a finite time in a solution of the non-collision type (for it is unknown whether there are such singularities of the non-collision type). Donald Saari showed in 1977 that for (or fewer particles) the set of initial states for which no global solution exists has the measure zero. He also proved that the set of initial conditions that lead to collisions has zero for every measure. It remains to show that the set of initial conditions with the singularities of the non-collision type has zero for the measure. The problem is unsolved (2016). If it were proven that the set of initial conditions that lead to singularities in Newton's theory of gravity has the measure zero, it would be shown in the mathematical sense "for almost all" initial conditions that Newton's theory of gravity is deterministic. ${\displaystyle N=3}$ ${\displaystyle N\geq 5}$ ${\displaystyle N=4}$ ${\displaystyle N=4}$ ${\displaystyle N}$ ${\displaystyle N\geq 5}$

• Problem 1B: The problem posed by Simon, which asks about the existence of singularities of the non-collision type for certain , is, however, as mentioned, solved by Xia, except for . ${\displaystyle N}$ ${\displaystyle N=4}$

The quantum mechanical version of the problem is solved, however, since according to Tosio Kato (1951) the Schrödinger equation with Coulomb potential has global solutions. But Simon mentions the most important still open problem in the area of ​​solvability questions for the Schrödinger equation, Konrad Jörgens' conjecture : Let on and be a finite union of closed submanifolds of . Furthermore, let it be essentially self-adjoint to (whereby the set of smooth functions with compact support) and is restricted from below. Then it is essentially self-adjoint . ${\displaystyle W(x)\geq V(x)}$ ${\displaystyle \mathbb {R} ^{\nu }}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{\nu }}$ ${\displaystyle -\Delta +V}$ ${\displaystyle C_{0}^{\infty }(\mathbb {R} ^{\nu }\setminus M)}$ ${\displaystyle C_{0}^{\infty }}$ ${\displaystyle -\Delta +W}$ ${\displaystyle C_{0}^{\infty }(\mathbb {R} ^{\nu }\setminus M)}$

Problem 2: Ergodic Theory

The ergodic theory is linked to the fundamentals of statistical mechanics. In thermodynamic equilibrium, macroscopic systems only depend on a few parameters, which indicates a uniform distribution in phase space. One model investigated for this purpose is the gas of hard spheres, for which Jakow Sinai demonstrated ergodicity in the 1960s.

• Problem 2A: Extend the Sinai proof to soft core potentials (continuous, repulsive potentials).

For potentials with attractive components, the approach to equilibrium according to Arthur Wightman could be explained by the fact that only part of the dynamics is ergodic, but at the limit of infinite volume (with finite particle density) it fills the entire phase space.

• Problem 2B: Verify Wightman's scenario for suitable potentials or otherwise show the approach to equilibrium.

Finally, Simon considers quantum theory and gives as an example a problem about the quantum mechanical Heisenberg model on an infinite lattice. Prove (problem 2C) that this is asymptotically Abelian (or refute it).

Problem 3: Long-term behavior of dynamic systems

Formulate a comprehensive theory of the long-term behavior of dynamic systems including the formation of turbulence . Simon himself admits that the formulation is very general, but justifies this with the importance of the area, which in his opinion has not yet reached a stage of maturity at this point in time (1984), so it is not yet clear what it really is crucial open questions are. He also saw progress with the question of the formation of turbulence ( Jean-Pierre Eckmann , David Ruelle ), but not with the theory of fully developed turbulence. Simon also found the connection to the theory of the Navier-Stokes equations on turbulence insufficiently clarified and the theory of the existence of solutions to the Navier-Stokes equation unsatisfactory. The latter problem is one of the Millennium Problems .

Problem 4: transport theory

• Problem 4A: Simon asks for a mechanical model in which Fourier's law of heat conduction follows on a microscopic basis. The system has the expansion and the temperature difference between the ends, then according to Fourier the heat output should be for . Simon notes that there has to be a mechanism for diffusion due to the interaction of the particles with one another, since non-interacting particles show no dependence of the heat conduction on (if the number of particles is increased when the length is increased so that the density remains constant). The problem is open. ${\displaystyle L}$ ${\displaystyle \Delta T}$ ${\displaystyle {\dot {Q}}\sim {\frac {1}{L}}}$ ${\displaystyle L\to \infty }$ ${\displaystyle L}$ ${\displaystyle L}$

• Problem 4B: Simon demands a strict justification of the cube formula in quantum statistics.

Problem 5: Heisenberg model

Models with closest neighbors of interacting spins on the lattice of the dimension are considered , where the spin on the unit sphere is in -dimensional space, which corresponds to the Ising model , the XY model (planar rotator) and the Heisenberg model . Another parameter in these models of statistical mechanics is the inverse temperature . The model is said to have long-range order if the expected value . The behavior of the models differs greatly according to the spatial dimension and topology of the spin space (which is also discrete in the Ising model and continuous in the XY and Heisenberg model). ${\displaystyle \nu }$ ${\displaystyle \sigma }$ ${\displaystyle D}$ ${\displaystyle D=1}$ ${\displaystyle D=2}$ ${\displaystyle D=3}$ ${\displaystyle \beta }$ ${\displaystyle \lim _{r\to \infty }\langle \sigma (0)\cdot \sigma (r)\rangle \neq 0}$

• Ising model ( ). For there is long-range order for sufficiently large (sufficiently low temperature) according to Rudolf Peierls . ${\displaystyle D=1}$ ${\displaystyle \nu \geq 2}$ ${\displaystyle \beta }$
• For there is no long-range order with continuous symmetries as in the Heisenberg model ( Mermin-Wagner theorem ) ${\displaystyle \nu =2}$
• There is a long-range order for for and large ( Jürg Fröhlich , Thomas C. Spencer , B. Simon 1976). ${\displaystyle \nu \geq 3}$ ${\displaystyle D\geq 2}$ ${\displaystyle \beta }$

Linked to this is the behavior of the spin-spin correlation function for large distances : does the function decay according to a power law or exponentially? For Fröhlich and Spencer showed that it only decays according to a power law for large and two dimensions. Exponential decay is expected for and two dimensions. ${\displaystyle \langle \sigma (0)\cdot \sigma (r)\rangle }$ ${\displaystyle r}$ ${\displaystyle D=2}$ ${\displaystyle \beta }$ ${\displaystyle D\geq 3}$

• Problem 5A: A strict proof is required in the case (Heisenberg model). ${\displaystyle D=3}$

Further problems concern the phase structure in the Heisenberg model.

• Problem 5B: It has to be proven that the equilibrium phases for low temperatures in the Heisenberg model are described by a single unit vector which, for example, indicates the direction of magnetization.

• Problem 5C: Requires a proof of the Griffiths-Kelly-Sherman (GKS) inequalities in the Heisenberg model for expectation values ​​of products of functions that are built up from spin-spin correlation functions. It was proven for the Ising model by Robert Griffiths (1967), DJ Kelly and S. Sherman (1968), for it was proven by Jean Ginibre in 1970, the case is open. ${\displaystyle D=2}$ ${\displaystyle D=3}$

• Problem 5D: Concerning the quantum version of the Heisenberg model. Prove that this has a long-range order for and sufficiently large (and thus a phase transition). ${\displaystyle \nu \geq 3}$ ${\displaystyle \beta }$

For the quantum mechanical problem, the existence of phases of long-range order for finite temperatures and the Heisenberg antiferromagnets (spin 1, 3/2, ...) with closest-neighbor interaction on a cubic lattice in three and more dimensions was suggested by Freeman Dyson , Elliott Lieb and Barry Simon proved it in 1978 (and for other spin systems such as the XY model with spin 1/2). The spin 1/2 case for the antiferromagnet and the case of the ferromagnet remained open. For the ground state (T = 0) E. Jordao Neves and J. Fernando Perez proved in 1986 the existence of long-range order for the two-dimensional Heisenberg antiferromagnet and spin what by Tom Kennedy, S. Shastry and Elliott Lieb on all dimensions greater than two and all Spins has been expanded. The case of the ferromagnet for finite temperature and three or more dimensions (Problem 5D) and the case of the antiferromagnet in the ground state for spin 1/2 and two dimensions remained open. ${\displaystyle S\geq {\tfrac {3}{2}}}$ ${\displaystyle S>0}$

Simon lists six other problems related to grid models.

Problem 6: Ferromagnetism

In ferromagnetism, there is a strong tendency for the electrons to align their spins in parallel, which, according to Werner Heisenberg, is explained by the fact that, because of the electron repulsion, the spatial wave function of the electrons is as antisymmetric as possible and, according to the Pauli principle , the spin wave function is as symmetrical as possible (parallel spins). Simon poses the problem of supporting this Heisenberg explanation mathematically in a realistic model. In 1962 , Elliott Lieb and Daniel Mattis had shown in one dimension that ferromagnetism cannot arise in this way (the total angular momentum of the ground state of an even number of electrons is zero in one dimension).

Problem 7: phase transitions in the continuum

This is about the proof of the existence of phase transitions in models with transition to the continuum (in contrast to lattice models) with somewhat realistic interaction. The phase transition is defined as a discontinuity in the free energy. The transition to the continuum corresponds to the limit of infinite volume with the density kept constant. Certain requirements are placed on the pair potential, which for example apply to the Lennard-Jones potential , which is often considered for interatomic interaction . In particular:

• Stability (for the sum of the particle pairs and particles), with a positive constant . ${\displaystyle \sum _{i<j}v_{ij}\geq -CN}$ ${\displaystyle N}$ ${\displaystyle C}$
• Growth condition ${\displaystyle |v(x)|\leq {\frac {C}{{(1+|x|)}^{3+\epsilon }}}}$

Problem 8: Strict theory of renormalization group

It deals with the theory of the renormalization group by Kenneth Wilson , which in some cases (nonlinear mapping of the unit interval according to Mitchell Feigenbaum , Jean-Pierre Eckmann, Collet , Oscar Lanford ) was already treated strictly mathematically, in its original application in statistical mechanics However, functions with an infinite number of variables appear and Simon poses the problem of finding a mathematically precise formulation using the example of the Ising model in dimensions. ${\displaystyle D}$

In addition, the proof of universality in the three-dimensional Ising model is presented as problem 8B (independence of the critical exponents from the relative relationships of the strength of the interaction between nearest neighbors in all three spatial directions).

The problem is open (even for the two-dimensional model, universality and conformal invariance in the scaling limit case was only strictly proven in 2012 by Stanislaw Smirnow and Dmitri Sergejewitsch Tschelkak ).

Problem 9: Asymptotic completeness of scattering processes

The problem area concerns the scattering theory, i.e. the study of the time-dependent Schrödinger equation for times . For a long time, progress in the strictly mathematical description was limited to up to three particles (three-body problem by Ludwig Faddejew and others in the 1960s). Significant progress in the many-particle case was achieved in 1978 by Volker Enß (methods of microlocal analysis), in 1981 by Eric Mourre (method of local positive commutators) and the geometric formulation of the problem by Simon, Schmuel Agmon , Volker Enß, Israel Michael Sigal and others. a. with separation of the movement of the center of gravity systems of the stable particle clusters and their inner movement and is the main problem of the mathematical scattering theory. Asymptotic completeness concerns the complete description of the possible scattering states for free particles and stable clusters (bound states) of particles. In Problem 9A, Simon presented the proof of asymptotic completeness for short-range potentials of the interaction of two particles in three or more dimensions and in Problem 9B for Coulomb potentials in three dimensions. ${\displaystyle t\to \pm \infty }$ ${\displaystyle t\to \pm \infty }$ ${\displaystyle V_{ij}}$

The problem for short-range potentials ( , ) in the -particle system was solved by Israel Michael Sigal and Avy Soffer in 1987, for long-range potentials in 1993 by Sigal and Soffer ( ) and Jan Derezinski . ${\displaystyle V_{ij}(r)\sim {\frac {1}{r^{\mu }}}}$ ${\displaystyle \mu >1}$ ${\displaystyle N}$ ${\displaystyle \mu \geq {\sqrt {3}}-1}$

Problem 10: quantum theory of atoms and molecules (Coulomb potentials)

The first problem concerns the total binding energy in an atom or molecule with electrons and nuclear charge (s) , whereby the nuclei are fixed ( Born-Oppenheimer approximation ). The ionization energy is . ${\displaystyle E(N,Z)}$ ${\displaystyle N}$ ${\displaystyle Z}$ ${\displaystyle \Delta E(N,Z)=E(N-1,Z)-E(N,Z)}$

• Problem 10A: Monotonicity of the ionization energy. . This corresponds to the fact that it takes more energy to remove the inner electrons than the outer ones. ${\displaystyle \Delta (N-1,Z)\geq \Delta (N,Z)}$

• Problem 10B: Asks for a rigorous proof of Scott's correction. This problem concerns the binding energy of a neutral atom in the Thomas-Fermi model . is developed for large . According to Elliott Lieb and Simon is in leading order in (with the Thomas Fermi energy ) and after is the term of next order (with a constant found by JMC Scott in 1952 ). ${\displaystyle E(Z,Z)=E(Z)}$ ${\displaystyle Z}$ ${\displaystyle \lim _{Z\to \infty }E(Z)=e_{TF}\cdot Z^{\frac {7}{3}}}$ ${\displaystyle Z}$ ${\displaystyle e_{TF}}$ ${\displaystyle e_{Scott}Z^{2}}$ ${\displaystyle e_{Scott}}$

• Problem 10C: Requires a corresponding asymptotic expansion (for large ones ) for the ionization energy . ${\displaystyle Z}$ ${\displaystyle \Delta E(Z,Z)}$

• Problem 10D: Simon asks about the maximum ionic charge of an atom (maximum number of electrons bound by an atom of the nuclear charge ). The existence of such a maximum charge has been proven by Mary Beth Ruskai and IM Sigal, for large it is believed to be asymptotic . What is required is strict evidence. ${\displaystyle N(Z)}$ ${\displaystyle Z}$ ${\displaystyle N(Z)}$ ${\displaystyle Z}$ ${\displaystyle N(Z)=Z}$

The last problem is linked to the evidence of "stability of matter" for fermionic matter according to Freeman Dyson and Andrew Lenard (1967) and later Elliott Lieb and Walter Thirring (1975). Instead, consider positively and negatively charged bosons (bosonic "protons" and "electrons", both of finite mass) and their binding energy . Is known (with constants ). In contrast to fermionic matter, bosonic matter was not stable. The question was, with which exponents the binding energy of the particles depended on ( , or a value in between). ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle E_{B}(N)}$ ${\displaystyle -DN^{\frac {5}{3}}\leq E_{B}(N)\leq -CN^{\frac {7}{5}}}$ ${\displaystyle C,D}$ ${\displaystyle \alpha ={\frac {7}{5}}}$ ${\displaystyle \alpha ={\frac {5}{3}}}$

• Problem 10E: Simon asks about barriers

${\displaystyle -AN^{\alpha }\leq E_{B}(N)\leq -BN^{\alpha }}$

with constants and , where is assumed. The barriers give a measure of the instability of boson matter. Corresponding bounds for "protons" (nuclei) of infinitely high mass were given by Elliott Lieb 1979. The problem was solved in 1988 by Lieb, Joseph Conlon and Horng-Tzer Yau ( is the correct exponent). ${\displaystyle A,B}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha ={\tfrac {7}{5}}}$ ${\displaystyle \alpha ={\tfrac {5}{3}}}$ ${\displaystyle {\frac {7}{5}}}$

Problem 11: existence of crystals

Most materials show a regular atomic arrangement in crystal lattices at sufficiently low temperatures, but there is no strict evidence of this in quantum mechanics. Prove that the basic state of an infinitely extended neutral system (edge ​​effects should not be included) of nuclei (atomic number ) and electrons tends towards a periodic limit value when the number of nuclei approaches infinity. ${\displaystyle Z}$

The problem is still largely open in the classical and quantum mechanical case (apart from the one-dimensional case).

Problem 12: Random and almost periodic potentials

The problem area concerns the Schrödinger equation with random or almost periodic potentials, as they occur in various problems in solid state physics.

Prototypes for random potentials are the Anderson model on a grid (which is used as a model for Anderson localization ) and the Fast Mathieu equation for almost periodic potential. The Anderson model in its discrete version is defined by the effect of the Hamilton operator on the wave function : ${\displaystyle u}$

${\displaystyle H_{\omega }u=\sum _{|j|=1}u(n+j)+V_{\omega }u(n)}$

Here is ( is the spatial dimension) and stands for a random variable with even distribution . The spectrum is and almost certainly (with probability 1) a dense point spectrum (localized states). The expectation is that this also applies for sufficiently large ones, but for small ones an interval with a purely absolutely continuous spectrum (extended wave function) and in the complement of in a dense point spectrum, where disappears the larger it becomes. In one dimension, localization (pure point spectrum) was first proven by I. Goldsheid, S. Molchanov and L. Pastur in 1977. Localization (in the Anderson model and similar models) for large coupling constants or energies near the edge of the spectrum has been proven in more than one dimension and there is the assumption that areas with a continuous spectrum exist for three and more dimensions (clearly for the wave function then enough space to avoid interference). The general expectation is that localization also applies to two dimensions, but not to three or more. ${\displaystyle n\in \mathbb {Z} ^{\nu }}$ ${\displaystyle \nu }$ ${\displaystyle \omega }$ ${\displaystyle [a,b]}$ ${\displaystyle [a-2\nu ,b+2\nu ]}$ ${\displaystyle \nu =1}$ ${\displaystyle |a-b|}$ ${\displaystyle \nu \geq 3}$ ${\displaystyle |b-a|}$ ${\displaystyle [c,d]\subset [a,b]}$ ${\displaystyle [c,d]}$ ${\displaystyle [a,b]}$ ${\displaystyle |c-d|}$ ${\displaystyle |a-b|}$

• Problem 12A: Evidence for and suitable (sufficiently small) values ​​of that the Anderson model has a purely absolutely continuous spectrum for a certain energy range. That is, there are extended (non-localized) states. Show that this does not apply to, but that only a dense point spectrum exists there. ${\displaystyle \nu \geq 3}$ ${\displaystyle b-a}$ ${\displaystyle \nu =2}$

• Problem 12B: Prove that in the Anderson model and in general for random potentials the transport has a diffusion character.
• Problem 12C: Prove that the integrated density of states at the mobility limit (the energy range in which the transition from the continuous to the discrete spectrum takes place) is constant in the energy. ${\displaystyle k(E)}$

Simon listed the problems again in the 2000 updated list.

Further problems concern the discrete almost periodic Mathieu operator:

${\displaystyle H_{\alpha ,\lambda ,\theta }u(n)=u(n+1)+u(n-1)+\lambda \cos(\pi \alpha n+\theta )u(n)}$

where the choice is mostly irrational, is the coupling constant and represents a phase. For rational , the spectrum is purely absolutely continuous. As became clear in the early 1980s, the spectrum depends not only on the coupling constant, but also on the arithmetic properties of . ${\displaystyle \alpha }$ ${\displaystyle \lambda }$ ${\displaystyle \theta \in [0,2\pi )}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

According to Peter Sarnak , the spectrum should depend in a certain way on the Diophantine properties of . For this purpose, Liouville numbers with good approximation properties are replaced by rational numbers and Roth numbers (named after Klaus Friedrich Roth ) with not so good properties. For a Roth number is for constants . ${\displaystyle \alpha }$ ${\displaystyle |\alpha -{\frac {p}{q}}|\geq {\frac {C}{q^{k}}}}$ ${\displaystyle C,k}$

• Problem 12D: We are looking for a confirmation of the following assumptions about the spectrum of the Fast Mathieu operator:
  • ${\displaystyle \alpha }$be a Liouville number and , then the spectrum is singularly continuous for almost all phases . ${\displaystyle \lambda \neq 0}$ ${\displaystyle \theta }$
  • ${\displaystyle \alpha }$be a Roth number and , then the spectrum is purely absolutely continuous for almost all phases. ${\displaystyle |\lambda |<2}$
  • ${\displaystyle \alpha }$be a Roth number and , then the spectrum is a dense point spectrum. ${\displaystyle |\lambda |>2}$
  • ${\displaystyle \alpha }$be a Roth number and , then the spectrum is purely singularly continuous and has the Lebesgue measure zero. ${\displaystyle |\lambda |=2}$

The (modified formulated) problem can be found in the updated list and has since been solved, for example Artur Avila proved in 2008 that (in the case of irrational ) the spectrum is absolutely continuous exactly if (where it is assumed). For the spectrum is almost certainly purely singularly continuous (B. Simon, Svetlana Jitomirskaya , Y. Gordon, Y. Last 1997) and according to Svetlana Jitomirskaya (1999) the spectrum is almost certainly a pure point spectrum (with which Anderson localization is present). is also called the critical value of the coupling constant. Avila and Jitomirskaya, for example, achieved results on the effects of the interaction of Diophantine properties of and coupling constant for the spectrum. ${\displaystyle \alpha }$ ${\displaystyle |\lambda |<2}$ ${\displaystyle \lambda \neq 0}$ ${\displaystyle |\lambda |=2}$ ${\displaystyle |\lambda |>2}$ ${\displaystyle \lambda =2}$ ${\displaystyle \alpha }$

• Problem 12E: Consider the continuous version of the Fast Mathieu operator:

${\displaystyle -{\frac {d^{2}}{dx^{2}}}+\lambda \cos(2\pi x)+\mu \cos(2\pi \alpha x+\theta )}$.

Show that this operator has a point spectrum for almost all phases and some values ​​of . ${\displaystyle \alpha ,\lambda ,\mu }$

Problem 13: self-avoiding random walks

As a model for the exact calculation of critical exponents (in connection with lattice spin models of field theory and applications with polymers), Simon considers self-avoiding paths on a lattice in dimensions. ${\displaystyle \phi ^{4}}$ ${\displaystyle \mathbb {Z} ^{d}}$ ${\displaystyle d}$

One of the interesting problems is the asymptotic behavior (for the number of steps ) of the number of self-avoiding paths of the step length . Another is that of the asymptotics of mean length with the critical exponent . When random walk would be , but you would expect for dimensions and for . Simon poses the problem of rigorously proving this. ${\displaystyle n\to \infty }$ ${\displaystyle c(n)}$ ${\displaystyle n}$ ${\displaystyle D(n)\sim c\cdot n^{\nu }}$ ${\displaystyle \nu }$ ${\displaystyle \nu ={\tfrac {1}{2}}}$ ${\displaystyle \nu \geq {\tfrac {1}{2}}}$ ${\displaystyle d\geq 4}$ ${\displaystyle \nu >{\tfrac {1}{2}}}$ ${\displaystyle d\leq 3}$

Numerical calculations support the conjecture (and suggest specifically for , and for , ). The case has been proven, the cases are open. ${\displaystyle d=2}$ ${\displaystyle \nu ={\tfrac {3}{4}}}$ ${\displaystyle d=3}$ ${\displaystyle \nu =0.5888}$ ${\displaystyle d\geq 5}$ ${\displaystyle d=2,3,4}$

With regard to the scaling limit (transition from the lattice to the continuum) was 2004, a significant advance is achieved by demonstrating that he a Schramm-Löwner evolution with equivalent. The existence of the scaling limit and its conforming invariance is open, however. ${\displaystyle \kappa ={\tfrac {3}{8}}}$

Problem 14: quantum field theory

• Problem 14A: Give a mathematically strict construction of quantum chromodynamics (QCD).

The QCD is an example of a renormalizable QFT. Since the construction of the QCD due to the fact that it represents a non-Abelian gauge field theory with fermions and is possibly too difficult:

• Problem 14B: Give a mathematically strict construction of a non-trivial renormalisable QFT (however, it should not show too simple UV behavior and not belong to the super renormalisable QFT), for example within the framework of constructive QFT .

It is implied that the usual four space-time dimensions are considered. Since, according to Simon, the majority of high-energy physicists assume that quantum electrodynamics (QED), although in very good agreement with experiments at relatively low energies, is not a consistent theory for high energies (existence of Landau-Poland):

• Problem 14C: Prove that QED is not a consistent theory.

In the standard model, the QED is embedded in the electroweak theory, a non-Abelian gauge theory, which often shows a different behavior due to asymptotic freedom . The existence of Landau poles (divergence of the coupling constant at finite energy) is connected with the question of the quantum triviality of the theory: the renormalized charge disappears - whereby the picture is often used that it is completely shielded by vacuum fluctuations - and thus corresponds to one "trivial" theory of free (non-interacting) particles. At higher energies (smaller distances), however, the bare charge is less and less shielded and finally diverges at the Landau pole.

An example of questions of consistency in QFT is the proof of the triviality of -QFT in dimensions by Jürg Fröhlich and Michael Aizenman 1981 ${\displaystyle \phi ^{4}}$ ${\displaystyle d>4}$

• Problem 14D: Prove that the QFT is inconsistent in four spacetime dimensions. ${\displaystyle \phi ^{4}}$

The triviality (vanishing renormalized coupling constant) was already assumed by Kenneth Wilson and John Kogut in 1974. Scalar field theories would be of practical importance for Higgs bosons, which in the Standard Model are embedded in non-Belian gauge theories. Since Simon's paper there have been advances in the non-perturbation-theoretical treatment of quantum field theories on the grid, also with regard to the triviality of scalar field theories in four dimensions. Strict evidence is still missing.

Problem 15: Cosmic Censorship

As a conclusion, Simon chooses a problem from the general theory of relativity (GTR), in which, according to the proofs of Stephen Hawking and Roger Penrose in the 1960s, singularities inevitably occur ( singularity theorem ) and not a relic of solutions with particularly high levels, as was previously assumed Symmetries were like some coordinate singularities in the Schwarzschild solution . According to the hypothesis of the cosmic censor by Roger Penrose, these are shielded from the rest of the universe in the ART by event horizons , there are no naked singularities . However, there are also opponents of the Cosmic Censorship Hypothesis and that was even the subject of a bet between Stephen Hawking (supporter of Cosmic Censorship) and Kip Thorne and John Preskill , who theoretically consider naked singularities in the ART to be possible.

Demetrios Christodoulou showed in the 1990s that naked singularities in GTR with scalar fields as matter can form under certain circumstances, but that these are unstable.

List from 2000

The problems concern Schrödinger operators and mainly comprise two problem areas, random potentials as they occur in quantum transport and the associated anomalous behavior of the spectra, and in the second (and in Simon view more difficult) problem area Coulomb potentials.

First, Schrödinger operators with ergodic (random) potentials and almost periodic potentials are considered as in problem 12 of the first list. Prototypes for random potentials are the Anderson model on a grid and the Fast Mathieu equation. The following problems are posed for the Anderson model:

• Problem 1: expanded states. Evidence for and suitable (sufficiently small) values ​​of that the Anderson model has a purely absolutely continuous spectrum for a certain energy range. That is, there are extended (non-localized) states. This corresponds to problem 12A in the original 1984 listing. ${\displaystyle \nu \geq 3}$ ${\displaystyle b-a}$
• Problem 2: Localization in two dimensions. Prove that for the spectrum of the Anderson model has a dense pure point spectrum for all values ​​of . In physics this corresponds to the Anderson localization . This was also mentioned in the original list as problem 12A. ${\displaystyle \nu =2}$ ${\displaystyle b-a}$
• Problem 3: Quantum Diffusion. Prove that for and values ​​of in which an absolutely continuous spectrum exists, the sum grows as if . This means that one has an expectation value , as is to be expected with diffusion. This corresponds to problem 12B in the original list (which is formulated a little differently there). ${\displaystyle \nu \geq 3}$ ${\displaystyle |b-a|}$ ${\displaystyle \sum _{n\in Z^{\nu }}n^{2}\cdot {|e^{(itH)}(n,0)|}^{2}}$ ${\displaystyle ct}$ ${\displaystyle t\to \infty }$ ${\displaystyle {\sqrt {\langle x(t)^{2}\rangle }}\sim {\sqrt {t}}}$

The prototype for almost periodic potentials is the almost periodic Mathieu operator, for which Simon formulates the following problems, all of which have now been solved:

• Problem 4: The Ten Martini Problem (by Mark Kac ). Proof for all and all irrational that the spectrum of the Hamilton operator (which is independent of) is a Cantor set , that is, it is nowhere dense. The ten martini problem was solved by Artur Avila and Svetlana Jitomirskaya . Joaquim Puig and Simon did the preliminary work themselves with Jean Bellissard . ${\displaystyle \lambda \neq 0}$ ${\displaystyle \alpha }$ ${\displaystyle \theta }$

• Problem 5: Prove that for all irrational and the spectrum of the Fast Mathieu operator the Lebesgue measure has zero. Solved in 2003 by Artur Avila and R. Krikorian . The case corresponds to the butterfly fractal by Douglas Hofstadter (who examined it in his dissertation in 1975 and had already suspected that it has Lebesgue measure zero). ${\displaystyle \alpha }$ ${\displaystyle \lambda =2}$

• Problem 6: Prove that the spectrum is for all irrational and purely absolutely continuous. Proven by Artur Avila. Problems 5 and 6 were already addressed in the original list (problem 12D, even if at that time the irrational character was distinguished from even finer). ${\displaystyle \alpha }$ ${\displaystyle \lambda <2}$ ${\displaystyle \alpha }$

The next problems deal with slowly decaying potentials.

• Issue 7: Is there potential for having a so a singular continuous spectrum has? The problem was solved positively by SA Denissov, and completely by Alexander Kiselev . ${\displaystyle V(x)}$ ${\displaystyle [0,\infty )}$ ${\displaystyle |V(x)|\leq C|x|^{(-1/2-\epsilon )}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle -{\frac {d^{2}}{dx^{2}}}+V}$

• Problem 8: Be a function with . Prove that has an absolutely continuous spectrum, with infinite multiplicity on and if . Proven for by Percy Deift and Rowan Killip. ${\displaystyle V}$ ${\displaystyle \mathbb {R} ^{\nu }}$ ${\displaystyle \int |x|^{(-\nu +1)}{|V(x)|}^{2}d^{\nu }x<\infty }$ ${\displaystyle -\Delta +V}$ ${\displaystyle [0,\infty )}$ ${\displaystyle \nu >2}$ ${\displaystyle \nu =1}$

The next problems concern the Schrödinger equation with Coulomb potential and especially the understanding of binding energies of atoms and molecules.

• Problem 9: Let the ground state energy of electrons in an atom with a nuclear charge and the smallest for that . Prove that for is bounded (another guess is that the difference is either zero or one). This corresponds to problem 10D of the original list. ${\displaystyle E(N,Z)}$ ${\displaystyle N}$ ${\displaystyle Z}$ ${\displaystyle N_{0}(Z)}$ ${\displaystyle N}$ ${\displaystyle E(N+j)=E(N)}$ ${\displaystyle N_{0}(Z)-Z}$ ${\displaystyle Z\to \infty }$
• Problem 10: What is the asymptotics of the ionization energy for ? The problem corresponds to 10C in the original listing. A related problem is the asymptotic behavior of the atomic radius. ${\displaystyle (\delta E)(Z)}$ ${\displaystyle Z\to \infty }$

The following problems also apply to atomic and molecular physics (and problem 13 to solid state physics), but are formulated more vaguely according to Simon:

• Problem 11: Give a mathematically meaningful formulation and strict justification of the shell model of the atoms.
• Problem 12: Can current "ab intio" techniques for determining molecular configurations in quantum chemistry be justified mathematically? We are looking for a mathematically strict way to get from the fundamental quantum mechanical formulation to the configurations of macromolecules.
• Problem 13 corresponds to problem 11 of the original list (existence of crystals)

Simon lists two other problems:

• Problem 14: Prove that the integrated density of states continuously depends on the energy (in one dimension and for the discrete case, the continuous dependence is proven, the higher-dimensional case is sought). For the Anderson model, this was problem 12C in the original listing. For the almost periodic math equation (with irrational ) Artur Avila and David Damanik proved that the integrated density of states is absolutely continuous if and only if (non-critical coupling constant). ${\displaystyle k(E)}$ ${\displaystyle \alpha }$ ${\displaystyle |\lambda |\neq 2}$
• Problem 15: Prove Elliott Lieb and Walter Thirring's conjecture about their constants for and . ${\displaystyle L_{(\gamma ,\nu )}}$ ${\displaystyle \nu =1}$ ${\displaystyle {\frac {1}{2}}<\gamma <{\frac {3}{2}}}$

Web links

• Eric Weisstein: Simon's Problems (according to the updated list from 2000)
• Simon's homepage (with online access to the article on his problems)

Individual evidence

1. ↑ Simon: Fifteen problems in mathematical physics, Oberwolfach Anniversary Volume, 1984, 423–454
2. ↑ Simon: Schrodinger Operators in the twentieth-first century, in: A. Fokas, A. Grigoryan, T. Kibble, B. Zegarlinski (eds.): Mathematical Physics 2000, Imperial College Press, London, 283-288
3. ↑ In the list of problems updated in 2000, Simon states that five have since been resolved.
4. ↑ Summary of the state of the problem according to John Baez, Struggles with the continuum , Arxiv 2016
5. ↑ The problem is not the collisions of two bodies that can be regularized, but rather "higher order" collisions
6. ↑ Simon complains, however, that no strict proof has been published except for particles and proof sketches for ${\displaystyle N=2}$ ${\displaystyle N=3,4,5}$
7. ^ F. Bonetto, JL Lebowitz, L. Rey-Bellet: Fourier's law, a challenge to theorists, Arxiv, 2000
8. ^ A. Dhar, Surprises in the theory of heat conduction, 2011 (pdf)
9. ↑ Fröhlich, Simon, Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Comm. Math. Phys., Vol. 50, 1976, pp. 79-95, online
10. ↑ Dyson, Lieb, Simon, Phase transition in quantum spin systems with isotropic and nonisotropic interactions, J. Stat. Phys. Volume 18, 1978, p. 335
11. ↑ In contrast to the classical case, in the quantum mechanical case there is a big mathematical difference between antiferromagnet and ferromagnet. For example: Lieb, Long range order for the quantum Heisenberg model ( Memento of the original from May 25, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. , 1999 
12. ↑ E. Jordao Neves, J. Fernando Perez, long range order in the ground state of two-dimensional anti-ferromagnet, Phys. Lett. A, Vol. 114, 1986, pp. 331-333
13. ↑ T. Kennedy, EH Lieb, S. Shastry, J. Stat. Phys., Vol. 53, 1988, pp. 1019-1030
14. ^ Lieb, Mattis, Theory of ferromagnetism and the ordering of electronic energy levels. In: Physical Review. Vol. 125, 1962, pp. 164-172
15. ↑ The proof in a somewhat artificial model was provided by David Ruelle in 1971, Existence of Phase Transitions in a Continuous Classical System , Phys. Rev. Lett., Vol. 27, 1971, p. 1040
16. ↑ IM Sigal, Asymptotic Completeness, AMS Translations 175, 1996, pp. 183–201 (pdf)
17. ^ Sigal, Soffer, The N-Particle scattering problem: asymptotic completeness for short range quantum systems, Annals of Mathematics, Volume 125, 1987, pp. 35-108
18. ^ Sigal, Soffer, Asymptotic completeness for N-particle long range scattering, Journal of the Am. Math. Soc., Volume 7, 1994, pp. 307-334, pdf
19. ↑ Derezinski, Asymptotic completeness of N-particle long range quantum systems, Annals of Math., Volume 138, 1993, pp. 427-476
20. ^ Dyson, Lenard Stability of matter, Part 1, J. Math. Phys., Volume 8, 1967, pp. 423-434, Volume 9, 1968, pp. 698-711
21. ^ Lieb, Thirring, Bound for the Kinetic Energy of Fermions which Proves the Stability of Matter, Phys. Rev. Lett., Vol. 35, 1975, pp. 687-689
22. ↑ The upper bound comes from Freeman Dyson, Ground state energy of a finite system of charged particles, J. Math. Phys., Volume 8, 1967, p. 1538, the lower from Dyson and Lenard, Stability of matter 1,2, J. Math. Phys., Vol. 8, 1967, 423-434, Vol. 9, 1968, pp. 698-711
23. ^ Lieb, The Law for Bosons, Phys. Lett. A, Volume 70, 1979, p. 71 ${\displaystyle N^{\tfrac {5}{3}}}$
24. ↑ JG Conlon, EH Lieb, H.-T. Yau: The law for charged bosons, Commun. Math. Phys. Vol. 116, 1988, pp. 417-448, Project Euclid ${\displaystyle N^{\frac {7}{5}}}$
25. ↑ Xavier Blanc, Mathieu Lewin, The Crystallization Conjecture: A Review, EMS Surveys in Math., 2015, Arxiv
26. ↑ Goldsheid, Molchanov, Pastur, A pure point spectrum for the stochastic one dimensional Schrödinger equation, Funct. Analysis Applic., Volume 11, 1977, pp. 1-10, see also Simon, Schrödinger operators in the 21st century, J. Math. Phys., Volume 41, 2000, pp. 3523-3555, Chapter VII (Ergodic Potentials ).
27. ↑ J. Fröhlich, T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys., Vol. 88, 1983, pp. 151-184
28. ↑ M. Aizenman, S. Molchanov, Localization at large disorder and extreme energies: an elementary derivation, Comm. Math. Phys., Vol. 157, 1993, pp. 235-277
29. ^ For example, Reed, Simon, Methods of Modern Mathematical Physics, Volume 4, Academic Press 1978
30. ^ Sarnak, Spectral behavior of quasiperiodic potentials, Comm. Math. Phys., Vol. 84, 1982, pp. 377-401, online
31. ↑ Some of the conjectures were wrong in the original Form 12D, as Yoram Last proved in his dissertation. Jitomirskaya, in: Fritz Gesztesy, From Mathematical Physics to Analysis, A walk in Barry Simon's Mathematical Garden II, Notices AMS, September 2016, p. 881
32. ↑ Avila, The absolutely continuous spectrum of the almost Mathieu operator , 2008. That a purely continuous spectrum existed for some was already known in advance and it was suspected for all . ${\displaystyle \alpha }$ ${\displaystyle \alpha }$
33. ^ AY Gordon, S. Jitomirskaya, Y. Last, B. Simon, Duality and singular continuous spectrum in the almost Mathieu equation, Acta Math., Volume 178, 1997, pp. 169-183. This shows that for those where the sequence of the whole numbers of the continued fraction expansion is not restricted (which is true for almost all ) there is a singular continuous spectrum for almost all phases. ${\displaystyle \alpha }$ ${\displaystyle \alpha }$
34. ↑ S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math., Vol. 150, 1999, pp. 1159-1175
35. ↑ Among others Artur Avila, Jiangong You, Qui Zhou, Sharp Phase transitions for the almost Mathieu operator, Arxiv 2015
36. ↑ According to Simon, two well-known mathematicians had bet against him that there was no point spectrum at all
37. ↑ For one can also simply show that ${\displaystyle d=1}$ ${\displaystyle \nu =1}$
38. ↑ David Brydges, Thomas Spencer, Self-avoiding walk in 5 and more dimensions, Comm. Math. Phys., Vol. 97, 1985, pp. 125-148, online
39. ↑ Madras, Slade, The self avoiding walk, Birkhäuser 1996
40. ^ Gordon Slade, The self avoiding walk, 2010, pdf
41. ↑ Lawler, Schramm, Werner, On the scaling limit of planar self-avoiding walk, 2004
42. ↑ These only have a finite number of divergent Feynman diagrams
43. ↑ For example Espriu, Tarrach, Ambiguities in QED: Renormalons versus Triviality, Phys. Lett. B, Vol. 383, 1996, pp. 482-486, Arxiv
44. ↑ See for example Ulli Wolff, Triviality of four dimensional phi ^ 4 theory on the lattice, Scholarpedia 2014
45. ^ Wilson, Kogut, The Renormalization Group and the -Expansion, Physics Reports, Vol. 12, 1974, p. 75 ${\displaystyle \epsilon }$
46. ↑ For example, Kuti, Shen, Supercomputing the effective action, Phys. Rev. Lett., Vol. 60, 1988, p. 85, Drummond, Duane, Horgan, Stochastic quantization simulation of φ4 theory, Nucl. Phys. B, Vol. 280, 1987, pp. 25-44
47. ↑ Hawking, Preskill and Thorne 1997 renewal of the bet , Caltech
48. ↑ Avila, Smitomirskaya, The Ten Martini Problem, Annals of Mathematics, Volume 170, 2009, pp. 303-340
49. ^ Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., Volume 244, 2004, pp. 297-309, Arxiv 2003
50. ↑ Bellissard, Simon, Cantor spectrum for the almost Mathieu equation, J. Funct. Anal., Vol. 48, 1982, pp. 408-419
51. ↑ Avila, Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schrodinger cocycles, Annals of Mathematics, Volume 164, 2006, pp. 911-940, Arxiv
52. ^ Avila, The absolutely continuous spectrum of the almost Mathieu operator, Arxiv 2008
53. ↑ SA Denisov, On the Coexistence of Absolutely Continuous and Continuous Singular Components of the Spectral Measure for Some Sturm-Liouville operator with Square summable potential. "J. Diff. Eq., Vol 191, 2003, pp 90-104.
54. ↑ A. Kiselev, Imbedded Singular Continuous Spectrum for Schrödinger Operators, J. of the AMS, Volume 18, 2005, pp 571-603, Arxiv 2001
55. ↑ Deift, Killip, Comm. Math. Phys., Volume 203, 1999, p. 341
56. ↑ Avila, Damanik, Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling, Inv. Math., Volume 172, 2008, pp. 439-453, Arxiv
57. ^ Lieb, Thirring, in Lieb, Simon, Wightman, Studies in Mathematical Physics, Princeton UP 1976