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Simon problems
Fifteen problems in mathematical physic

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 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.

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Context

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

  • H f ( N ) {\displaystyle {\mathcal {H}}_{f}^{(N)}} is the space of functions on L 2 ( R 3 N ; C 2 N ) {\displaystyle L^{2}(\mathbb {R} ^{3N};\mathbb {C} ^{2N})} which are asymmetrical under exchange of the N {\displaystyle N} spin and space coordinates.12 Equivalently, the subspace of ( L 2 ( R 3 ) ⊗ C 2 ) ⊗ N {\displaystyle (L^{2}(\mathbb {R} ^{3})\otimes \mathbb {C} ^{2})^{\otimes N}} which is asymmetrical under the exchange of the N {\displaystyle N} factors.
  • The Hamiltonian is H ( N , Z ) := ∑ i = 1 N ( − Δ i − Z | x i | ) + ∑ i < j 1 | x i − x j | {\displaystyle H(N,Z):=\sum _{i=1}^{N}(-\Delta _{i}-{\frac {Z}{|x_{i}|}})+\sum _{i<j}{\frac {1}{|x_{i}-x_{j}|}}} . Here x i ∈ R 3 {\displaystyle x_{i}\in \mathbb {R} ^{3}} is the coordinate of the i {\displaystyle i} -th particle, Δ i {\displaystyle \Delta _{i}} is the Laplacian with respect to the coordinate x i {\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 E ( N , Z ) := min H f H ( N , Z ) {\displaystyle E(N,Z):=\min _{{\mathcal {H}}_{f}}H(N,Z)} , that is, the ground state energy of the ( N , Z ) {\displaystyle (N,Z)} system.
  • We define N 0 ( Z ) {\displaystyle N_{0}(Z)} to be the smallest value of N {\displaystyle N} such that E ( N + j , Z ) = E ( N , Z ) {\displaystyle E(N+j,Z)=E(N,Z)} for all positive integers j {\displaystyle j} ; it is known that such a number always exists and is always between Z {\displaystyle Z} and 2 Z {\displaystyle 2Z} , inclusive.13

The 1984 list

Simon listed the following problems in 1984:14

No.Short nameStatementStatusYear 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 1984.15[needs update] In 1977, Saari showed that this is true for 4-body problems.16?
(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.1718

In 1991, Gerver showed that 3n-body problems in the plane for some sufficiently large value of n also undergo non-collisional singularities.19

1989
2nd(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.[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.20

?
(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.[needs update]?
(c) Asymptotic abelianness for the quantum Heisenberg dynamicsProve or disprove that the multidimensional quantum Heisenberg model is asymptotically abelian.Open as of 1984.[needs update]?
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.[needs update]?
4th(a) Fourier's heat lawFind a mechanical model in which a system of size L {\displaystyle L} with temperature difference Δ T {\displaystyle \Delta T} between its ends has a rate of heat temperature that goes as L − 1 {\displaystyle L^{-1}} in the limit L → ∞ {\displaystyle L\to \infty } .Open as of 1984.[needs update]?
(b) Kubo's formulaJustify 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 v = 2 {\displaystyle v=2} 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.[needs update]?
(b) Pure phases and low temperatures for the v ≥ 3 {\displaystyle v\geq 3} classical Heisenberg modelProve that, in the D = 3 {\displaystyle D=3} model at large beta and at dimension v ≥ 3 {\displaystyle v\geq 3} , the equilibrium states form a single orbit under S O ( 3 ) {\displaystyle SO(3)} : the sphere.
(c) GKS for classical Heisenberg modelsLet f {\displaystyle f} and g {\displaystyle g} be finite products of the form ( σ α ⋅ σ γ ) {\displaystyle (\sigma _{\alpha }\cdot \sigma _{\gamma })} in the D = 3 {\displaystyle D=3} model. Is it true that < f g > Λ , β ≥< f > Λ , β < g > Λ , β {\displaystyle <fg>_{\Lambda ,\beta }\geq <f>_{\Lambda ,\beta }<g>_{\Lambda ,\beta }}  ?
(d) Phase transitions in the quantum Heisenberg modelProve that for v ≥ 3 {\displaystyle v\geq 3} 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.[needs update]?
7thExistence of continuum phase transitionsShow that for suitable choices of pair potential and density, the free energy is non- C 1 {\displaystyle C^{1}} at some beta.Open as of 1984.[needs update]?
8th(a) Formulation of the renormalization groupDevelop mathematically precise renormalization transformations for v {\displaystyle v} -dimensional Ising-type systems.Open as of 1984.[needs update]?
(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 that ⊕   Ran   Ω a + = L 2 ( X ) {\displaystyle \oplus ~{\text{Ran}}~\Omega _{a}^{+}=L^{2}(X)} .Open as of 1984.21[needs update]?
(b) Asymptotic completeness for Coulomb potentialsSuppose v = 3 , V i j ( x ) = e i j | x | − 1 {\displaystyle v=3,V_{ij}(x)=e_{ij}|x|^{-1}} . Prove that ⊕   Ran   Ω a D , + = L 2 ( X ) {\displaystyle \oplus ~{\text{Ran}}~\Omega _{a}^{D,+}=L^{2}(X)} .
10th(a) Monotonicity of ionization energy(a) Prove that ( Δ E ) ( N − 1 , Z ) ≥ ( Δ E ) ( N , Z ) {\displaystyle (\Delta E)(N-1,Z)\geq (\Delta E)(N,Z)} .Open as of 1984.[needs update]?
(b) The Scott correctionProve that lim Z → ∞ ( E ( Z , Z ) − e T F Z 7 / 3 ) / Z 2 {\displaystyle \lim _{Z\to \infty }(E(Z,Z)-e_{TF}Z^{7/3})/Z^{2}} exists and is the constant found by Scott.
(c) Asymptotic ionizationFind the leading asymptotics of ( Δ E ) ( Z , Z ) {\displaystyle (\Delta E)(Z,Z)} .
(d) Asymptotics of maximal ionized chargeProve that lim Z → ∞ N ( Z ) / Z = 1 {\displaystyle \lim _{Z\to \infty }N(Z)/Z=1} .
(e) Rate of collapse of Bose matterFind suitable C 1 , C 2 , α {\displaystyle C_{1},C_{2},\alpha } such that − C 1 N α ≤ E ~ B ( N , N ; 1 ) ≤ C 2 N α {\displaystyle -C_{1}N^{\alpha }\leq {\tilde {E}}_{B}(N,N;1)\leq C_{2}N^{\alpha }} .
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.[needs update]?
12th(a) Existence of extended states in the Anderson modelProve that in v ≥ 3 {\displaystyle v\geq 3} and for small λ {\displaystyle \lambda } that there is a region of absolutely continuous spectrum of the Anderson model, and determine whether this is false for v = 2 {\displaystyle v=2} .Open as of 1984.[needs update]?
(b) Diffusive bound on "transport" in random potentialsProve that Exp ( δ 0 , ( e i t H N → e − i t H ) 2 δ 0 ) ≤ c ( 1 + | t | ) {\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.
(c) Smoothness of k ( E ) {\displaystyle k(E)} through the mobility edge in the Anderson modelIs k ( E ) {\displaystyle k(E)} , the integrated density of states, a C ∞ {\displaystyle C^{\infty }} function in the Anderson model at all couplings?
(d) Analysis of the almost Mathieu equationVerify the following for the almost Mathieu equation:
  • If α {\displaystyle \alpha } is a Liouville number and λ ≠ 0 {\displaystyle \lambda \neq 0} , then the spectrum is purely singular continuous for almost all θ {\displaystyle \theta } .
  • If α {\displaystyle \alpha } is a Roth number and | λ | < 2 {\displaystyle |\lambda |<2} , then the spectrum is purely absolutely continuous for almost all θ {\displaystyle \theta } .
  • If α {\displaystyle \alpha } is a Roth number and | λ | > 2 {\displaystyle |\lambda |>2} , then the spectrum is purely dense pure point.
  • If α {\displaystyle \alpha } is a Roth number and | λ | = 2 {\displaystyle |\lambda |=2} , then σ ( h ) {\displaystyle \sigma (h)} has Lebesgue measure zero and the spectrum is purely singular continuous.
(e) Point spectrum in a continuous almost periodic modelShow that − d 2 d x 2 + λ cos ⁡ ( 2 π x ) + μ cos ⁡ ( 2 π α x + θ ) {\displaystyle -{\frac {d^{2}}{dx^{2}}}+\lambda \cos(2\pi x)+\mu \cos(2\pi \alpha x+\theta )} has some point spectrum for suitable α , λ , μ {\displaystyle \alpha ,\lambda ,\mu } and almost all θ {\displaystyle \theta } .
13thCritical exponent for self-avoiding walksLet D ( n ) {\displaystyle D(n)} be the mean displacement of a random self-avoiding walk of length n {\displaystyle n} . Show that v := lim n → ∞ n − 1 ln ⁡ D ( n ) {\displaystyle v:=\lim _{n\to \infty }n^{-1}\ln D(n)} is 1 2 {\displaystyle {\frac {1}{2}}} for dimension at least four and is greater otherwise.Open as of 1984.[needs update]?
14th(a) Construct QCDGive a precise mathematical construction of quantum chromodynamics.Open as of 1984.[needs update]?
(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 φ 4 4 {\displaystyle \varphi _{4}^{4}} Prove that a nontrivial φ 4 4 {\displaystyle \varphi _{4}^{4}} theory does not exist.
15thCosmic censorshipFormulate 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.22

The 2000 list

The Simon problems as listed in 2000 (with original categorizations), are:2324

No.Short nameStatementStatusYear solved
Quantum transport and anomalous spectral behavior
1stExtended statesProve that the Anderson model has purely absolutely continuous spectrum for v ≥ 3 {\displaystyle v\geq 3} and suitable values of b − a {\displaystyle b-a} in some energy range.??
2ndLocalization in 2 dimensionsProve that the spectrum of the Anderson model for v = 2 {\displaystyle v=2} is dense pure point.??
3rdQuantum diffusionProve that, for v ≥ 3 {\displaystyle v\geq 3} and values of | b − a | {\displaystyle |b-a|} where there is absolutely continuous spectrum, that ∑ n ∈ Z ν n 2 | e i t H ( n , 0 ) | 2 {\displaystyle \sum _{n\in \mathbb {Z} ^{\nu }}n^{2}|e^{itH}(n,0)|^{2}} grows like c t {\displaystyle ct} as t → ∞ {\displaystyle t\to \infty } .??
4thTen Martini problemProve that the spectrum of h α , λ , θ {\displaystyle h_{\alpha ,\lambda ,\theta }} is a Cantor set (that is, nowhere dense) for all λ ≠ 0 {\displaystyle \lambda \neq 0} and all irrational α {\displaystyle \alpha } .Solved by Puig (2003).25262003
5thProve that the spectrum of h α , λ , θ {\displaystyle h_{\alpha ,\lambda ,\theta }} has measure zero for λ = 2 {\displaystyle \lambda =2} and all irrational α {\displaystyle \alpha } .Solved by Avila and Krikorian (2003).27282003
6thProve that the spectrum of h α , λ , θ {\displaystyle h_{\alpha ,\lambda ,\theta }} is absolutely continuous for λ = 2 {\displaystyle \lambda =2} and all irrational α {\displaystyle \alpha } .??
7thDo there exist potentials V ( x ) {\displaystyle V(x)} on [ 0 , ∞ ) {\displaystyle [0,\infty )} such that | V ( x ) | ≤ C | x | 1 2 + ε {\displaystyle |V(x)|\leq C|x|^{{\frac {1}{2}}+\varepsilon }} for some ε {\displaystyle \varepsilon } and such that − d 2 d x 2 + V {\displaystyle -{\frac {d^{2}}{dx^{2}}}+V} has some singular continuous spectrum?Essentially solved by Denisov (2003) with only L 2 {\displaystyle L^{2}} decay.

Solved entirely by Kiselev (2005).293031

2003, 2005
8thSuppose that V ( x ) {\displaystyle V(x)} is a function on R ν {\displaystyle \mathbb {R} ^{\nu }} such that ∫ | x | − ν + 1 | V ( x ) | 2 d ν x < ∞ {\displaystyle \int |x|^{-\nu +1}|V(x)|^{2}d^{\nu }x<\infty } , where ν ≥ 2 {\displaystyle \nu \geq 2} . Prove that − Δ + V {\displaystyle -\Delta +V} has absolutely continuous spectrum of infinite multiplicity on [ 0 , ∞ ) {\displaystyle [0,\infty )} .??
Coulomb energies
9thProve that N 0 ( Z ) − Z {\displaystyle N_{0}(Z)-Z} is bounded for Z → ∞ {\displaystyle Z\to \infty } .??
10thWhat are the asymptotics of ( δ E ) ( Z ) := E ( Z , Z − 1 ) − E ( Z , Z ) {\displaystyle (\delta E)(Z):=E(Z,Z-1)-E(Z,Z)} for Z → ∞ {\displaystyle Z\to \infty } ???
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 problems
14thProve that the integrated density of states k ( E ) {\displaystyle k(E)} is continuous in the energy.| k(E1 + ΔE) - k(E1) | < ε?
15thLieb-Thirring conjectureProve the Lieb-Thirring conjecture on the constants L γ , ν {\displaystyle L_{\gamma ,\nu }} where ν = 1 , 1 2 < γ < 3 2 {\displaystyle \nu =1,{\frac {1}{2}}<\gamma <{\frac {3}{2}}} .??

See also

References

  1. Simon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3. 978-1-86094-230-3 ↩

  2. Marx, C. A.; Jitomirskaya, S. (2017). "Dynamics and Spectral Theory of Quasi-Periodic Schrödinger-type Operators". Ergodic Theory and Dynamical Systems. 37 (8): 2353–2393. arXiv:1503.05740. doi:10.1017/etds.2016.16. S2CID 119317111. /wiki/ArXiv_(identifier) ↩

  3. Damanik, David. "Dynamics of SL(2,R)-Cocycles and Applications to Spectral Theory; Lecture 1: Barry Simon's 21st Century Problems" (PDF). Beijing International Center for Mathematical Research, Peking University. Retrieved 2018-07-07. http://www.math.pku.edu.cn/teachers/gansb/conference09/Damanik-Minicourse-2009-08-09.pdf ↩

  4. Simon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3. 978-1-86094-230-3 ↩

  5. "Simon's Problem" (PDF). University of Colorado Boulder. https://home.cs.colorado.edu/~alko5368/lecturesCSCI3090/lecture11.pdf ↩

  6. "Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Retrieved 2018-07-07. http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12 ↩

  7. Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Retrieved 2018-07-07. https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani ↩

  8. Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Retrieved 2018-07-07. https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani ↩

  9. Tao, Terry (2014-08-12). "Avila, Bhargava, Hairer, Mirzakhani". What's New. Retrieved 2018-07-07. /wiki/Terence_Tao ↩

  10. Simon, Barry (1984). "Fifteen problems in mathematical physics". Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (PDF). Birkhäuser. pp. 423–454. Retrieved 24 June 2021. http://www.math.caltech.edu/SimonPapers/R27.pdf ↩

  11. Coley, Alan A. (2017). "Open problems in mathematical physics". Physica Scripta. 92 (9): 093003. arXiv:1710.02105. Bibcode:2017PhyS...92i3003C. doi:10.1088/1402-4896/aa83c1. S2CID 3892374. /wiki/ArXiv_(identifier) ↩

  12. Simon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3. 978-1-86094-230-3 ↩

  13. Simon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3. 978-1-86094-230-3 ↩

  14. Simon, Barry (1984). "Fifteen problems in mathematical physics". Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (PDF). Birkhäuser. pp. 423–454. Retrieved 24 June 2021. http://www.math.caltech.edu/SimonPapers/R27.pdf ↩

  15. Simon, Barry (1984). "Fifteen problems in mathematical physics". Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (PDF). Birkhäuser. pp. 423–454. Retrieved 24 June 2021. http://www.math.caltech.edu/SimonPapers/R27.pdf ↩

  16. Saari, Donald G. (October 1977). "A global existence theorem for the four-body problem of Newtonian mechanics". Journal of Differential Equations. 26 (1): 80–111. Bibcode:1977JDE....26...80S. doi:10.1016/0022-0396(77)90100-0. https://doi.org/10.1016%2F0022-0396%2877%2990100-0 ↩

  17. Xia, Zhihong (1992). "The Existence of Noncollision Singularities in Newtonian Systems". Annals of Mathematics. 135 (3): 411–468. doi:10.2307/2946572. JSTOR 2946572. MR 1166640. /wiki/Doi_(identifier) ↩

  18. Saari, Donald G.; Xia, Zhihong (April 1995). "Off to infinity in finite time" (PDF). Notices of the American Mathematical Society. 42 (5): 538–546. https://www.ams.org/journals/notices/199505/saari-2.pdf ↩

  19. Gerver, Joseph L (January 1991). "The existence of pseudocollisions in the plane". Journal of Differential Equations. 89 (1): 1–68. Bibcode:1991JDE....89....1G. doi:10.1016/0022-0396(91)90110-U. https://doi.org/10.1016%2F0022-0396%2891%2990110-U ↩

  20. Simon, Barry (1984). "Fifteen problems in mathematical physics". Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (PDF). Birkhäuser. pp. 423–454. Retrieved 24 June 2021. http://www.math.caltech.edu/SimonPapers/R27.pdf ↩

  21. Simon, Barry (1984). "Fifteen problems in mathematical physics". Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (PDF). Birkhäuser. pp. 423–454. Retrieved 24 June 2021. http://www.math.caltech.edu/SimonPapers/R27.pdf ↩

  22. Simon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3. 978-1-86094-230-3 ↩

  23. Simon, Barry (2000). "Schrödinger Operators in the Twenty-First Century". Mathematical Physics 2000. Imperial College London. pp. 283–288. doi:10.1142/9781848160224_0014. ISBN 978-1-86094-230-3. 978-1-86094-230-3 ↩

  24. Weisstein, Eric W. "Simon's Problems". mathworld.wolfram.com. Retrieved 2021-06-22. https://mathworld.wolfram.com/SimonsProblems.html ↩

  25. Weisstein, Eric W. "Simon's Problems". mathworld.wolfram.com. Retrieved 2021-06-22. https://mathworld.wolfram.com/SimonsProblems.html ↩

  26. Puig, Joaquim (1 January 2004). "Cantor Spectrum for the Almost Mathieu Operator". Communications in Mathematical Physics. 244 (2): 297–309. arXiv:math-ph/0309004. Bibcode:2004CMaPh.244..297P. doi:10.1007/s00220-003-0977-3. S2CID 120589515. /wiki/ArXiv_(identifier) ↩

  27. Weisstein, Eric W. "Simon's Problems". mathworld.wolfram.com. Retrieved 2021-06-22. https://mathworld.wolfram.com/SimonsProblems.html ↩

  28. Ávila Cordeiro de Melo, Artur; Krikorian, Raphaël (1 November 2006). "Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles". Annals of Mathematics. 164 (3): 911–940. arXiv:math/0306382. doi:10.4007/annals.2006.164.911. S2CID 14625584. /wiki/ArXiv_(identifier) ↩

  29. Weisstein, Eric W. "Simon's Problems". mathworld.wolfram.com. Retrieved 2021-06-22. https://mathworld.wolfram.com/SimonsProblems.html ↩

  30. Denisov, Sergey A. (June 2003). "On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm–Liouville operators with square summable potential". Journal of Differential Equations. 191 (1): 90–104. Bibcode:2003JDE...191...90D. doi:10.1016/S0022-0396(02)00145-6. https://doi.org/10.1016%2FS0022-0396%2802%2900145-6 ↩

  31. Kiselev, Alexander (27 April 2005). "Imbedded singular continuous spectrum for Schrödinger operators". Journal of the American Mathematical Society. 18 (3): 571–603. doi:10.1090/S0894-0347-05-00489-3. https://doi.org/10.1090%2FS0894-0347-05-00489-3 ↩