The authors consider linear ordinary differential equations on the 2-torus with coefficient matrix in the special linear group from the 2-torus with two frequencies. The results concern reducibility fully or ‘almost’ to a constant coefficient or a rotations case by a linear coordinate change, which in the periodic case is given always possible by Floquet theory. It is well known that reducibility depends on the properties of the (irrational) frequency ratio. Here the scaling is such that the frequency vector is of the form \((\alpha, 1)\) and \(\alpha\) is referred to as the frequency.
This reducibility problem has a direct connection to the spectral theory of Schrödinger operators with quasi-periodic potential, in particular the occurrence of absolutely continuous spectrum. The introduction of the present paper discusses this and the historical background with relations to KAM theory and previous results, notably by E. I. Dinaburg and Ya. G. Sinai [Funct. Anal. Appl. 9, 279–289 (1976); translation from Funkts. Anal. Prilozh. 9, No. 4, 8–21 (1975; Zbl 0333.34014)] and L. H. Eliasson [Commun. Math. Phys. 146, No. 3, 447–482 (1992; Zbl 0753.34055)].
Some implications of the main results for the Schrödinger equations as formulated by the authors are: for sufficiently large energy or analytically small potential, the Schrödinger system is either uniformly hyperbolic or has zero Lyapunov exponent, and for generic small real analytic potentials the spectrum is a Cantor set.
The first main result concerns non-perturbative reducibility in the sense that smallness assumptions are independent of Diophantine constants of the irrational frequency. It ensures analytic reducibility for coefficients analytic in a strip with respect to the phases and whose phase dependent part is sufficiently small with bound independent of the rational or Diophantine rotation number w.r.t. the frequency. Here “rotation number” refers to that of the fundamental matrix of the system.
The second result yields “almost reducibility”, a normal form sense of reducibility: roughly speaking there is a sequence of coordinate changes which brings the system closer and closer to constant. The result is analogous to the previous one when removing the assumption on the rotation number and replacing “analytically reducible” by “almost reducible”.
Next, the concept of reducibility is replaced by “rotations reducible”, meaning that the conjugated system has rotation matrix coefficients. The analogous result to the first theorem replaces reducible by rotations reducible and requires the rotation number of the system to be Diophantine w.r.t. the frequency. The bound is the same as in the second theorem.
In case of Liouvillean frequency, the authors prove the following depending on \(\beta(\alpha) = \limsup_{n\to\infty} (\ln q_n )/q_n\), with \(p_n /q_n\) the continued fraction expansion of \(\alpha\in(0,1)\). The second theorem can be strengthened to imply reducible instead of almost reducible, but for \(\beta(\alpha) = 0\) the bound in the result depends on \(\beta(\alpha)\).
The proofs build on the aforementioned [Zbl 0753.34055], relying on KAM type iterations and adding Floquet theory. As an application the Schrödinger case is considered and it is shown that all abstract results apply under suitable energy bounds.