Abstract
We prove that the quasi-periodic Schrödinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if the frequency is Diophantine and the potential is sufficiently small in the corresponding \(C^k\) topology. This extends the work of Eliasson [19] and Avila–Jitomirskaya [5] from the analytic topology to the finitely differentiable one which is much broader, revealing the interesting phenomenon that small oscillation of the potential leads to both zero Lyapunov exponent in the whole spectrum and purely absolutely continuous spectrum. Our result is based on a refined quantitative \(C^{k,k_0}\) almost reducibility theorem which only requires a quite low initial regularity “\(k>14\tau +2\)” and much of the regularity “\(k_0\le k-2\tau -2\)” is conserved in the end, where \(\tau \) is the Diophantine constant of the frequency.
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Acknowledgements
The author is deeply grateful to the referees for their careful review of this paper so that its readability is greatly improved in various perspectives. The author would like to thank Jiangong You and Qi Zhou for useful discussions at Chern Institute of Mathematics, and is grateful to Pedro Duarte for his persistent support at University of Lisbon as well as to Silvius Klein for his consistent support from PUC-Rio. This work is supported by FAPERJ Programa Pós-Doutorado Nota 10-2021, PTDC/MAT-PUR/29126/2017 and NSFC grant (11671192).
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Appendix
Appendix
Proof of Theorem 3.1
For readers who are quite familiar with the analytic KAM scheme, this proof can be skipped since the structure is similar to that in [13]. But the estimates here are sharp compared with those in [13] (see Remark 3.3), so we prefer to provide the detailed proof for self-containedness.
Recall that \(sl(2,{{\mathbb {R}}})\) is isomorphic to su(1, 1), which consists of matrices of the form
with \(t\in {{\mathbb {R}}}\), \(v\in {{\mathbb {C}}}\). The isomorphism between them is given by \(A\rightarrow MAM^{-1}\), where
and a simple calculation yields
where \(x,y,z\in {{\mathbb {R}}}\). SU(1, 1) is the corresponding Lie group of su(1, 1). We will prove this theorem in SU(1, 1), which is isomorphic to \(SL(2,{{\mathbb {R}}})\). \(\square \)
We distinguish two cases:
Non-resonant case For \(0<|n |\leqslant N=\frac{2}{r-r'} |\ln \epsilon |\), we have
by (3.2) with \(D>\frac{2}{\sigma }\), we have
It is well known that (5.1) and (5.2) are the conditions which are used to overcome the small denominator problem in KAM theory.
Define
Our goal is to solve the cohomological equation
i.e.,
Here \(\mathcal {T}_N\) is the truncation operator such that
Take the Fourier transform for (5.4) and compare the corresponding Fourier coefficients of the two sides. By (5.1) (apply it twice to solve the off-diagonal) along with (5.2) (apply it once to solve the diagonal), we obtain that if \(Y\in \Lambda _N\), then
which gives
Moreover, we have \(A^{-1}Y(\theta +\alpha )A \in \Lambda _N\) by (5.3). For \(\eta =\epsilon ^{3\sigma }\), we define \(\mathcal {B}_r^{nre}(\epsilon ^{3\sigma })\) by (3.1), then we have \(\Lambda _N \subset \mathcal {B}_r^{nre}(\epsilon ^{3\sigma })\).
Since \(\epsilon ^{3\sigma }\geqslant 13\Vert A\Vert ^2\epsilon ^{\frac{1}{2}}\) (it holds by \(\sigma \) being smaller than \(\frac{1}{6}\) and \(\tilde{D}\) depending on \(\sigma \)), by Lemma 3.1 we have \(Y\in \mathcal {B}_r\) and \(f^{re}\in \mathcal {B}_r^{re}(\epsilon ^{3\sigma })\) such that
with \(|Y |_r\leqslant \epsilon ^{\frac{1}{2}}\) and
By (5.3)
and
Moreover, we can compute that
by (5.7), we have
Finally, if we denote
then we have
Resonant case In fact, we only need to consider the case in which A is elliptic with eigenvalues \(\{e^{i\rho },e^{-i\rho }\}\) for \(\rho \in {{\mathbb {R}}}\backslash \{0\}\) since if \(\rho \in i{{\mathbb {R}}}\), then the non-resonant condition is always satisfied due to the Diophantine condition on \(\alpha \) and then it actually belongs to the non-resonant case.
Claim 2
\(n_*\) is the unique resonant site with
Proof
Indeed, if there exists \(n_{*}^{'}\ne n_*\) satisfying \(|2\rho - \langle n_{*}^{'},\alpha \rangle |< \epsilon ^{\sigma }\), then by the Diophantine condition of \(\alpha \), we have
which implies that \(|n_{*}^{'} |>2^{-\frac{1}{\tau }}\kappa ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N> 2N^2.\) \(\square \)
Since we have
the smallness condition on \(\epsilon \) implies that
Thus,
which implies that
Then by Lemma 8.1 of Hou–You [22], one can find \(P\in SU(1,1)\) with
such that
Denote \(g=PfP^{-1}\), by (3.2) we have:
Now we define
For \(\eta =\epsilon ^{\sigma }\), we define the decomposition \(\mathcal {B}_r=\mathcal {B}_r^{nre}(\epsilon ^{\sigma }) \bigoplus \mathcal {B}_r^{re}(\epsilon ^{\sigma })\) as in (3.1) with A substituted by \(A'\). Recall that su(1, 1) consists of matrices of the form
with \(t\in {{\mathbb {R}}}\), \(v\in {{\mathbb {C}}}\). Direct computation shows that any \(Y\in \mathcal {B}_r^{nre}(\epsilon ^{\sigma })\) takes the precise form:
Since \(\epsilon ^{\sigma }\geqslant 13\Vert A'\Vert ^2 (\epsilon ') ^{\frac{1}{2}}\), we can apply Lemma 3.1 to remove all the non-resonant terms of g, which means there exist \(Y\in \mathcal {B}_r\) and \(g^{re}\in \mathcal {B}_r^{re}(\eta )\) such that
with \(|Y |_r\leqslant (\epsilon ')^{\frac{1}{2}}\) and \(|g^{re}|_r\leqslant 2\epsilon '\).
Combining with the Diophantine condition on the frequency \(\alpha \) and the Claim 2, we have:
Let \(N':=2^{-\frac{1}{\tau }}\kappa ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N\), then we can rewrite \(g^{re}(\theta )\) as
Define the \(4\pi {{\mathbb {Z}}}^d\)-periodic rotation \(Q(\theta )\) as below:
So we have
One can also show that
where
and
Moreover,
Now we return back from su(1, 1) to \(sl(2,{{\mathbb {R}}})\). Denote
then we have:
By (5.9) and (5.12), we have the following estimates:
By (5.23) and (5.24), direct computation shows that
It immediately implies that
Thus, we can rewrite (5.20) as
with
Now recall that Baker–Campbell–Hausdorff formula [28] says that
where \([X,Y]=XY-YX\) denotes the Lie Bracket and \(\cdots \) denotes the sum of higher order terms. Using this formula and by a simple calculation, (5.26) gives
where
and
By (5.8) and (5.23), we obtain \(|t|\leqslant \epsilon ^{\sigma }\) and
Finally, the following estimate is straightforward:
This finishes the proof of Theorem 3.1. \(\square \)
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Cai, A. The Absolutely Continuous Spectrum of Finitely Differentiable Quasi-Periodic Schrödinger Operators. Ann. Henri Poincaré 23, 4195–4226 (2022). https://doi.org/10.1007/s00023-022-01192-y
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DOI: https://doi.org/10.1007/s00023-022-01192-y