- Almost Mathieu operator
In
mathematical physics , the almost Mathieu operator arises in the study of thequantum Hall effect . It is given by: H^{lambda,alpha}_omega u] (n) = u(n+1) + u(n-1) + 2 lambda cos(2pi (omega + nalpha)) u(n), ,
acting as a
self-adjoint operator on the Hilbert space ell^2(mathbb{Z}).Here alpha,omega inmathbb{T}, lambda > 0 are parameters.In pure mathematics, its importance comes from the fact of being one of the best-understood examples of anergodic Schrödinger operator . For example, three problems (now all solved) ofBarry Simon 's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator. [Barry Simon, Schrödinger operators in the twenty-first century, Mathematical physics 2000, 283--288, Imp. Coll. Press, London, 2000.]For lambda = 1, the almost Mathieu operator is sometimes called Harper's equation.
The spectral type
If alpha is a rational number, then H^{lambda,alpha}_omegais a periodic operator and by
Floquet theory its spectrum is purelyabsolutely continuous.Now to the case when alpha is irrational.Since the transformation omega mapsto omega + alpha is minimal,it follows that the spectrum of H^{lambda,alpha}_omega does notdepend on omega . On the other hand, by ergodicity, the supports ofabsolutely continuous, singular continuous, and pure point parts of the spectrum arealmost surely independent of omega .It is now known, that
- For 0 < lambda < 1, H^{lambda,alpha}_omegahas surely purely absolutely continuous spectrum. [A. Avila, The absolutely continuous spectrum of the almost Mathieu operator, Preprint.] (This was one of Simon's problems.)
- For lambda= 1, H^{lambda,alpha}_omegahas almost surely purely singular continuous spectrum. [Gordon, A. Y.; Jitomirskaya, S.; Last, Y.; Simon, B. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178 (1997), no. 2, 169--183.] (It is not known whether eigenvalues can exist for exceptional parameters.)
- For lambda > 1, H^{lambda,alpha}_omegahas almost surely pure point spectrum and exhibits
Anderson localization . [Jitomirskaya, Svetlana Ya. Metal-insulator transition for the almost Mathieu operator. Ann. of Math. (2) 150 (1999), no. 3, 1159--1175.] (It is known that almost surely can not be replaced by surely.) [J. Avron and B. Simon, Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc. 6 (1982), 81--85.] [S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Comm. Math. Phys. 165 (1994), 201--205. ]
That the spectral measures are singular when lambda geq 1 follows (through the work of Last and Simon) [Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999).] from the lower bound on the
Lyapunov exponent gamma(E) given by: gamma(E) geq max {0,log(lambda)}. ,
This lower bound was proved independently by Avron, Simon and
Michael Herman , after an earlier almost rigorous argument of Aubry and André. In fact, when E belongs to the spectrum, the inequality becomes an equality (the Aubry-André formula), proved byJean Bourgain and Svetlana Jitomirskaya. [J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, Journal of Statistical Physics 108 (2002), 1203--1218.]The structure of the spectrum
Another striking characteristic of the almost Mathieu operator is that its spectrum is a
Cantor set for all irrational alpha and lambda > 0. This was shown by Avila and Jitomirskaya [A. Avila and S. Jitomirskaya, The Ten Martini problem, Preprint.] solving the by-then famous "Ten Martini Problem" (also one of Simon's problems) after several earlier results (including generically [J. Bellissard and B. Simon, Cantor spectrum for the almost Mathieu equation, J. Funct. Anal. 48 (1982), 408--419.] and almost surely [Puig, Joaquim Cantor spectrum for the almost Mathieu operator. Comm. Math. Phys. 244 (2004), no. 2, 297–309.] with respect to the parameters).Furthermore, the measure of the spectrum of the almost Mathieu operator is known to be
: Leb(sigma(H^{lambda,alpha}_omega)) = |4 - 4 lambda| ,
for all lambda > 0. For lambda = 1 this means that the spectrum has zero measure (this was first proposed by
Douglas Hofstadter and later became one of Simon's problems [ A. Avila and R. Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles, Annals of Mathematics 164 (2006), 911--940.] ). For lambda eq 1 , the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky.The study of the spectrum for lambda =1 leads to the
Hofstadter's butterfly , where the spectrum is shown as a set.References
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