Simon's Problems

A set of 15 open problems on Schrödinger operators proposed by mathematical physicist 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 nu>=3 and suitable values of b-a that the Anderson model has purely absolutely continuous spectrum in some energy range.

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

3. Quantum diffusion. Prove that for nu>=3 and values of |b-a| where there is a.c. spectrum that sum_(n in Z^nu)n^2|e^(itH)(n,0)|^2 grows as ct as t->infty.

4. Ten Martini problem. Prove for all lambda!=0 and all irrational alpha that spec(h_(alpha,lambda,theta)) (which is theta independent) is a Cantor set, that is, that it is nowhere dense.

5. Prove for all irrational alpha and lambda=2 that spec(h_(alpha,lambda,theta)) has measure zero.

6. Prove for all irrational alpha and lambda<2 that the spectrum is purely absolutely continuous.

7. Do there exist potentials V(x) on [0,infty) so that |V(x)|<=C|x|^(-1/2-epsilon) for some epsilon>0 and so that -(d^2)/(dx^2)+V has some singular continuous spectrum?

8. Let V be a function on R^nu which obeys


Prove that -Delta+V has a.c. spectrum of infinite multiplicity on [0,infty) if nu>=2.

9. Prove that N_0(Z)-Z is bounded as Z->infty.

10. What is the asymptotics of (deltaE)(Z) as Z->infty?

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, k(E), is continuous in the energy.

15. Prove the Lieb-Thirring conjecture on their constants L_(gamma,nu) for nu=1 and 1/2<gamma<3/2.

The ten martini problem (#4) was solved by Puig (2003).

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 L^2 and not a pointwise decay, and solved in its entirety by Kiselev.

See also

Schrödinger Operator

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Avila, A. and Krikorian, R. "Reducibility or Non-Uniform Hyperbolicity for Quasiperiodic Schrodinger Cocycles." Jun. 26, 2003., 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." Preprint., J. "Cantor Spectrum for the Almost Mathieu Operator. Corollaries of Localization, Reducibility and Duality." Sept. 1, 2003., B. "Schrödinger Operators in the Twenty-First Century." Feb 17, 2000.

Referenced on Wolfram|Alpha

Simon's Problems

Cite this as:

Weisstein, Eric W. "Simon's Problems." From MathWorld--A Wolfram Web Resource.

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