Simon's problems are a set of 15 problems on Schrödinger operators originally posed as open by Barry Simon (2000). This set of problems follows up a 1984 list of open problems in mathematical physics also proposed by Simon, of which thirteen involved Schrödinger operators.

1. Extended states. Prove for and suitable values of that the Anderson model has purely absolutely continuous spectrum in some energy range.

2. Localization in two dimensions. Prove that for , the spectrum of the Anderson model is dense pure point for all values of .

3. Quantum diffusion. Prove that for and values of where there is a.c. spectrum that grows as as .

4. Ten martini problem. Prove for all and all irrational that (which is independent) is a Cantor set, that is, that it is nowhere dense.

5. Prove for all irrational and that has measure zero.

6. Prove for all irrational and that the spectrum is purely absolutely continuous.

7. Do there exist potentials on so that for some and so that has some singular continuous spectrum?

8. Let be a function on which obeys

Prove that has a.c. spectrum of infinite multiplicity on if .

9. Prove that is bounded as .

10. What is the asymptotics of as ?

11. Make mathematical sense of the shell model of an atom.

12. Is there a mathematical sense in which one can justify from first principles current techniques for determining molecular configurations?

13. Prove that the ground state of some neutral system of molecules and electrons approaches a periodic limit as the number of nuclei goes to infinity.

14. Prove the integrated density of states, , is continuous in the energy.

15. Prove the Lieb-Thirring conjecture on their constants for and .

The ten martini theorem resolves problem #4 (Avila and Jitomirskaya 2009).

The conjecture on zero measure in the Anderson model equation (#5) was solved by Avila and Krikorian (2003).

Problem #7 was essentially solved by Denissov (2003), although he only has an and not a pointwise decay, and solved in its entirety by Kiselev.

Problem #8 is Simon's L² conjecture. Becker and Yang (2026) report a counterexample in every dimension .

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References

Avila, A. and Jitomirskaya, S. "The Ten Martini Problem." Ann. Math. 170, 303-342, 2009. https://doi.org/10.4007/annals.2009.170.303.Avila, A. and Krikorian, R. "Reducibility or Non-Uniform Hyperbolicity for Quasiperiodic Schrodinger Cocycles." Jun. 26, 2003. https://arxiv.org/abs/math/0306382.Becker, S. and Yang, M. "Lenses Make Many Points: Pure Point Spectrum Disproves Simon's -Conjecture." 22 Oct 2026. https://arxiv.org/abs/2610.31637.Denissov, S. A. "On the Coexistence of Absolutely Continuous and Singular Continuous Components of the Spectral Measure for Some Sturm-Liouville Operators with Square Summable Potential." J. Diff. Eq. 191, 90-104, 2003.Kiselev, A. "Imbedded Singular Continuous Spectrum for Schrödinger Operators." J. Amer. Math. Soc. 18, 571-603, 2005. https://doi.org/10.1090/s0894-0347-05-00489-3.Puig, J. "Cantor Spectrum for the Almost Mathieu Operator. Corollaries of Localization, Reducibility and Duality." Sept. 1, 2003. https://arxiv.org/abs/math-ph/0309004.Simon, B. "Schrödinger Operators in the Twenty-First Century." Feb 17, 2000. https://web.archive.org/web/20000525133550/http://www.ma.utexas.edu:80/mp_arc-bin/mpa?yn=00-78.

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Simon's Problems

Cite this as:

Weisstein, Eric W. "Simon's Problems." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SimonsProblems.html

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