The ten martini theorem states that the operator spectrum of the almost Mathieu operator 
is a Cantor
set for every irrational frequency 
and every nonzero real coupling 
(Avila and Jitomirskaya 2009). In the normalization
![(H_(lambda,alpha,theta)u)_n=u_(n+1)+u_(n-1)+2lambdacos[2pi(nalpha+theta)]u_n,](/images/equations/TenMartiniTheorem/NumberedEquation1.svg) |
the operator acts on square-summable doubly infinite sequences, and the phase 
is an arbitrary real number.
For irrational 
, the operator
spectrum is independent of 
.
Avila and Jitomirskaya (2009) proved the theorem, resolving the conjecture known as the ten martini problem. Barry Simon introduced this name after Mark Kac offered ten martinis to anyone who solved the problem (Simon 1982). The supercritical ten martini problem replaces the cosine potential by a more general real analytic function.
