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An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. (English) Zbl 0533.34023
We consider the stationary Schrödinger 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.

##### MSC:
 34L99 Ordinary differential operators 81Q15 Perturbation theories for operators and differential equations in quantum theory
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