an:03845994
Zbl 0533.34023
Moser, J??rgen; P??schel, J??rgen
An extension of a result by Dinaburg and Sinai on quasi-periodic potentials
EN
Comment. Math. Helv. 59, 39-85 (1984).
00149146
1984
j
34L99 81Q15
Schr??dinger equation; Floquet solutions; spectral gaps; quasi-periodic potential; rotation number
We consider the stationary Schr??dinger equation \((*)\quad Ly=- y''+qy=\lambda y\) on the real line, where q is a quasi-periodic potential with basic frequencies \(\omega =(\omega_ 1,...,\omega_ d)\). For such potentials the rotation number \(\alpha\) (\(\lambda)\) is well defined. The spectral gaps of L are precisely the intervals of constancy of \(\alpha\), and there, \(\alpha(\lambda)=(j,\omega)/2\) for some integer vector \(j=(j_ 1,...,j_ d)\) (''gap labelling'').
We suppose that \(\omega\) is Diophantine, and that q extends to a real analytic function on its hull. Then the following is proven. If \(\mu =(k,\omega)/2\) is sufficiently large and badly approximable by all other resonances \((j,\omega)\)/2, \(j\neq k\), then the spectral gap \([\alpha,\beta]=\alpha^{-1}(\mu)\) is generically open, and (*) has Floquet solutions \(e^{i\mu x}(\chi_ 1+x\chi_ 2)\), \(e^{i\mu x}\chi_ 2\) for \(\lambda =\alpha,\beta\). If the gap is collapsed, that is, if \(\alpha =\beta\), then all solutions are of the form \(e^{i\mu x}\chi\). The functions \(\chi\) are all quasiperiodic with frequencies \(\omega\) and extend to real analytic functions on their hull.
This complements a result of Dinaburg and Sinai who proved the existence of solutions of the second kind for \(\lambda =\alpha^{-1}(\mu)\) in case \(\mu\) is sufficiently large and badly approximable by all resonances \((j,\omega)\)/2. In fact, the points in the absolutely continuous spectrum provided by their theorem are cluster points of the spectral gaps constructed above, and their result can be recovered from our construction by a limiting process.