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