Almost Mathieu operator

In mathematical physics, the almost Mathieu operator arises in the study of the quantum 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 an ergodic Schrödinger operator. For example, three problems (now all solved) of Barry 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 $H^\{lambda,alpha\}\_omega$is 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 , $H^\{lambda,alpha\}\_omega$has 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 $lambda=\; 1$, $H^\{lambda,alpha\}\_omega$has 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 $lambda\; >\; 1$, $H^\{lambda,alpha\}\_omega$has 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 by Jean 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