Almost Mathieu operator: Difference between revisions

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: <math> \gamma(E) \geq \max \{0,\log(\lambda)\}. \, </math>

This lower bound was proved independently by Avron, Simon and [[Michael Herman (mathematician)|Michael Herman]], after an earlier almost rigorous argument of Serge Aubry and Gilles André. In fact, when <math> E </math> belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), proved by [[Jean Bourgain]] and [[Svetlana Jitomirskaya]].<ref>{{cite journal |first1=J. |last1=Bourgain |first2=S. |last2=Jitomirskaya |title=Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential |journal=[[Journal of Statistical Physics]] |volume=108 |year=2002 |issue=5–6 |pages=1203–1218 |doi=10.1023/A:1019751801035 |s2cid=14062549 }}</ref>

==The structure of the spectrum==