Википедия:Песочница: различия между версиями
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| Строка 693: | Строка 693: | ||
such that, for $0<\ep<\ep_0$, the perturbation series for the operator \eqref{eq_longrange_def} is convergent, and the operator $H(x)$ satisfies Anderson localization for $x\in \R\setminus(\Z+\omega\cdot\Z+1/2)$. |
such that, for $0<\ep<\ep_0$, the perturbation series for the operator \eqref{eq_longrange_def} is convergent, and the operator $H(x)$ satisfies Anderson localization for $x\in \R\setminus(\Z+\omega\cdot\Z+1/2)$. |
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\end{prop} |
\end{prop} |
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\section{Some properties of Schr\"odinger eigenvectors} |
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In this section, we will summarize some basic properties of the discrete Schr\"odinger eigenfunctions in bounded domains. These properties are either well known or straightforward. Let $C\subset \Z^d$. Denote the Laplace operator on $\ell^2(C)$ by |
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$$ |
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(\Delta_C\psi)(\bn)=\sum\limits_{\bm\in C\colon |\bm-\bn|_1=1}\psi(\bm). |
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$$ |
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Let $A,B\subset \Z^d$. We will define $\bn\in A\cup B$ to be {\it simply reachable from $B$} using the following recurrent definition: |
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\begin{enumerate} |
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\item Any point $\bn\in B$ is simply reachable. |
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\item Let $\bn\in A\cup B$, $\bm\in A$, $|\bm-\bn|=1$ and the set |
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$$ |
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\{\bm'\in A\cup B \colon |\bn-\bm'|\le 1,\,\bm'\neq \bm\} |
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$$ |
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is not empty and only consists of simply reachable points. Then we declare $\bm$ to be simply reachable. |
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\end{enumerate} |
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The above definition can be understood as follows. The eigenvalue equation |
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$$ |
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\sum\limits_{\bm\in A\cup B\colon |\bm-\bn|_1=1}\psi(\bm)=E\psi(\bn) |
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$$ |
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relates values of $\psi$ at different points of $A\cup B$. If $\psi(\bn)$ and all $\psi(\bm)$, except for one value $\bm=\bm_0$, are known, the equation lets us determine the remaining value $\psi(\bm_0)$. A simply reachable point, in other words, is a point $\bm\in A\cup B$ such that $\psi(\bm)$ can be determined from the values of $\psi$ on $B$ using a finite sequence of such operations. |
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We will say that $B$ satisfies {\it simple unique continuation property} (SUP) for $A$ if all points of $A$ are simply reachable from $B$. |
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\begin{lem} |
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\label{lemma_unique} |
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Suppose that $B$ satisfies SUP for $A$. There exists $\delta_0=\delta_0(A,B)>0$ independent of $V$ such that the following is true: for any $M>0$ there is $\delta_1=\delta_1(A,B,M)>0$, also independent of $V$, such that for any $\psi$ satisfying |
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\bee |
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\label{eq_unique_condition} |
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H_{A\cup B}\psi=E\psi,\quad \|\psi\|_{\ell^2(A\cup B)}=1;\quad |\psi(\bm)|<\delta_0\,\,\forall \bm\in B, |
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\ene |
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we have, for some $\bm\in A\cup B$, the following: |
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$$ |
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|(V_\bm-E)\psi(\bm)|\ge \delta_1,\quad |V_\bm-E|\ge M. |
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$$ |
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\end{lem} |
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\begin{rem} |
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For any fixed $V$ and large $M$, one cannot satisfy \eqref{eq_unique_condition}: if a function is small on $B$, then the continuation procedure will result in small values, with bounds that only depend on $B$. Therefore, $|V_{\bm}-E|$ has to be large for some $\bm$. Moreover, since the new values of $\psi$ are calculated using $|V_{\bm}-E|\psi(\bm)|$ rather than $|V_{\bm}-E|$, the former also cannot be too small. |
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Note that the constants can be chosen independent of $C$, since there are only finitely many possibilities (depending on $A$, $B$) that can possibly affect the results. |
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\end{rem} |
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\begin{proof} |
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The following argument can be easily made more rigorous. Let us start from $A\setminus B$ and gradually apply the eigenvalue equation to recover values of $\psi_{\bm}$ on $B$. At each step, we are obtaining a new value $\psi_{\bn}$ as a combination $\psi_{\bm}$ and $(V_\bm-E)\psi_{\bm}$ already obtained. The initial data is assumed to be small. However, we ultimately obtain an $\ell^2$-normalized eigenvector, and therefore, at some point, we obtain a value $\approx 1$. Consider the first time this happened. The only possibility for that is through $(V_\bm-E)\psi_{\bm}$, where $\psi_{\bm}$ was small, which precisely means that $|V_{\bm}-E|$ must be large. |
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\end{proof} |
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The following version, obtained by simple rescaling, will be important: |
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\begin{cor} |
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In the above result, replace $\Delta$ by $\varepsilon\Delta$ with $\varepsilon>0$. Then the conclusion can be restated as follows: |
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we have, for some $\bm\in A$, |
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$$ |
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|(V_\bm-E)\psi(\bm)|\ge \delta_1 \varepsilon,\quad |V_\bm-E|\ge M\varepsilon. |
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$$ |
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\end{cor} |
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\begin{rem} |
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In Lemma \ref{lemma_unique}, one can replace $\Delta_A$ by a weighted Laplacian with non-zero weights, with the constants depending on upper and lower bounds on weights. Clearly, using zero weights would violate unique continuation. However, the change of weights of the edges connecting vertices in $A\setminus B$ would not affect the result at all, even if we use zero weights, since, once the eigenfunction is fixed in $A\setminus B$, we are only use non-modified edges to perform unique continuation. |
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\end{rem} |
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\begin{rem} |
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\label{remark_extended_sup} |
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In Lemma \ref{lemma_unique}, one can replace $H_A$ by $H_{A'}$, where $A\subset A'$. Suppose that $B\subset A$ satisfies SUP for $H_{A'}$; in other words, any eigenfunction of $H_{A'}$ admits a unique continuation from $A\setminus B$ into $B$ (but not necessarily into $A'$). Then the statement still holds, and does not depend on the form of $H_{A'}$ outside $A$. The proof repeats without changes. |
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\end{rem} |
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We will also need some elementary information about eigenfunction decay, which follows from elementary perturbation theory. |
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\begin{prop} |
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\label{prop_clustering} |
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Let $A$ and $H_A$ as above, and $A=A_1\cup\ldots \cup A_n$ be a disjoint union. Assume that |
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$$ |
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|V_{\bm}-V_{\bn}|\ge\eta,\quad \bm\in A_j,\,\,\bn\in A_j\,\,j\neq k. |
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$$ |
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Then there exists an orthonormal basis $\{\psi_{\bm}\colon \bm\in A\}$ of eigenfunctions of $H_A$ such that if $\bm\in A_j$, then |
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$$ |
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|\psi_{\bm}(\bn)|\le C(A,\{A_j\})\eta_j^{-\dist(\bn,A_j)}. |
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$$ |
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\end{prop} |
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In other words, if the values of $V$ are clustered, then the corresponding eigenfunctions decay exponentially away from the clusters. The proof immediately follows from the perturbation theory. Note that the statement only makes sense for $\eta\gg 1$, otherwise one can absorb the bound into $C(A,\{A_j\})$. The constant does not depend on $V$ other than through $\eta$. In particular, it does not depend on the energy range inside of a cluster; however, it does depend on the number of lattice sites in each cluster through $C$. |
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\begin{rem} |
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In Proposition \ref{prop_clustering}, one can replace the Laplacian by a weighted Laplacian as long as the weights are bounded from above by $1$, with the same constants. One can also replace it by a long range operator if the distance function is modified accordingly. |
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\end{rem} |
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Finally, we will need the following property related to differentiation of eigenvalues with respect to a parameter. The following is often referred to as Hellmann--Feynman variational formula. |
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\begin{prop} |
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\label{prop_hell} |
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Let $A(t)$ be a family of self-adjoint matrices that depends on a parameter $t$ in a smooth way. Let $\lambda(t)$ be a branch of an isolated eigenvalue of $A(t)$, and let $\psi(t)$ be an $\ell^2$-normalized eigenvector. Then |
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\bee |
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\label{eq_hell} |
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\lambda'(t)=\frac{d}{dt}\<A(t)\psi(t),\psi(t)\>=\<A'(t)\psi(t),\psi(t)\>. |
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\ene |
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\end{prop} |
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\begin{proof} |
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The remaining term in the right hand side of \eqref{eq_hell} is $\lambda(t)\frac{d}{dt}\langle \psi(t),\psi(t)\rangle $, which is identically zero due to normalization of $\psi(t)$. |
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\end{proof} |
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In particular, if $A(t)=A_0+f(t)\<\be_j,\cdot\>\be_j$, then |
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$$ |
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\lambda'(t)=f'(t)\langle\be_j\psi(t),\be_j\rangle=f'(t)|\psi(t;j)|^2. |
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$$ |
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| Строка 791: | Строка 882: | ||
\section{Some properties of Schr\"odinger eigenvectors} |
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In this section, we will summarize some basic properties of the discrete Schr\"odinger eigenfunctions in bounded domains. These properties are either well known or straightforward. |
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Let $A\subset \Z^d$ be a bounded connected subset. We will also consider $A$ as a sub-graph which inherits all edges from $\Z^d$ that connect points on $A$. On $\ell^2(A)$, consider the following operator: |
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$$ |
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(H_A\psi)(\bn)=(\Delta_A \psi)(\bn)+V_{\bn}\psi(\bn)=\sum\limits_{\bm\in A\colon |\bm-\bn|_1=1}\psi(\bm)+V_{\bn}\psi(\bn), |
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$$ |
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where $V_{\bn}$ is a real-valued potential. In the following claims, one can come up with different, more or less straightforward generalizations, but we will limit ourselves to a class of cases which is still more general that we need. Let $B\subset A$ be a connected subset. Call a point of $\bm\in B$ {\it simply accessible} if there exists a point $\bn\in A$ such that $|\bn-\bm|_1=1$ and all points of $\bm'\in A$ such that $|\bn-\bm'|_1\le 1$, except maybe for $\bm$, are either in $A\setminus B$ or simply accessible. This definition is recurrent: we start from $A\setminus B$ and gradually determine points to be accessible. |
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We will say that $B\subset A$ satisfies {\it simple unique continuation property} (SUP) if all points of $B$ are simply accessible. In other words, SUP means that any eigenfunction of the Laplacian (or Schr\"odinger operator) is uniquely determined by its values on $A\setminus B$, and this determination can be performed by calculating its values at points of $B$, one by one, by using the eigenfunction equation. We will need the following lemma related to simple version of the unique continuation. |
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\begin{lem} |
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\label{lemma_unique} |
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Let $A$ and $H_A$ as above and suppose that $B\subset A$ satisfies SUP. There exists $\delta_0=\delta_0(A,B)>0$ independent of $V$ such that the following is true: for any $M>0$ there is $\delta_1=\delta_1(A,B,M)>0$, also independent of $V$, such that for any $\psi$ satisfying |
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$$ |
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H_A\psi=E\psi,\quad \|\psi\|_{\ell^2(A)}=1;\quad |\psi(\bm)|<\delta_0\,\,\forall \bm\in A\setminus B, |
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$$ |
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we have, for some $\bm\in A$, the following: |
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$$ |
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|(V_\bm-E)\psi(\bm)|\ge \delta_1,\quad |V_\bm-E|\ge M. |
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$$ |
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\end{lem} |
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\begin{rem} |
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For any fixed $V$ and large $M$, the concluding claim is clearly impossible. This corresponds to the fact that an $\ell^2$-normalized eigenfunction cannot vanish too much around the boundary. In other words, the only mechanism of extreme vanishing of eigenfunctions is to have very large jumps in the values of $V$. |
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\end{rem} |
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\begin{proof} |
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The following argument can be easily made more rigorous. Let us start from $A\setminus B$ and gradually apply the eigenvalue equation to recover values of $\psi_{\bm}$ on $B$. At each step, we are obtaining a new value $\psi_{\bn}$ as a combination $\psi_{\bm}$ and $(V_\bm-E)\psi_{\bm}$ already obtained. The initial data is assumed to be small. However, we ultimately obtain an $\ell^2$-normalized eigenvector, and therefore, at some point, we obtain a value $\approx 1$. Consider the first time this happened. The only possibility for that is through $(V_\bm-E)\psi_{\bm}$, where $\psi_{\bm}$ was small, which precisely means that $|V_{\bm}-E|$ must be large. |
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\end{proof} |
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The following version, obtained by simple rescaling, will be important: |
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\begin{cor} |
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In the above result, replace $\Delta$ by $\varepsilon\Delta$ with $\varepsilon>0$. Then the conclusion can be restated as follows: |
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we have, for some $\bm\in A$, |
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$$ |
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|(V_\bm-E)\psi(\bm)|\ge \delta_1 \varepsilon,\quad |V_\bm-E|\ge M\varepsilon. |
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$$ |
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\end{cor} |
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\begin{rem} |
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In Lemma \ref{lemma_unique}, one can replace $\Delta_A$ by a weighted Laplacian with non-zero weights, with the constants depending on upper and lower bounds on weights. Clearly, using zero weights would violate unique continuation. However, the change of weights of the edges connecting vertices in $A\setminus B$ would not affect the result at all, even if we use zero weights, since, once the eigenfunction is fixed in $A\setminus B$, we are only use non-modified edges to perform unique continuation. |
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\end{rem} |
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\begin{rem} |
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\label{remark_extended_sup} |
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In Lemma \ref{lemma_unique}, one can replace $H_A$ by $H_{A'}$, where $A\subset A'$. Suppose that $B\subset A$ satisfies SUP for $H_{A'}$; in other words, any eigenfunction of $H_{A'}$ admits a unique continuation from $A\setminus B$ into $B$ (but not necessarily into $A'$). Then the statement still holds, and does not depend on the form of $H_{A'}$ outside $A$. The proof repeats without changes. |
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\end{rem} |
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We will also need some elementary information about eigenfunction decay, which follows from elementary perturbation theory. |
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\begin{prop} |
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\label{prop_clustering} |
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Let $A$ and $H_A$ as above, and $A=A_1\cup\ldots \cup A_n$ be a disjoint union. Assume that |
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$$ |
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|V_{\bm}-V_{\bn}|\ge\eta,\quad \bm\in A_j,\,\,\bn\in A_j\,\,j\neq k. |
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$$ |
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Then there exists an orthonormal basis $\{\psi_{\bm}\colon \bm\in A\}$ of eigenfunctions of $H_A$ such that if $\bm\in A_j$, then |
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$$ |
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|\psi_{\bm}(\bn)|\le C(A,\{A_j\})\eta_j^{-\dist(\bn,A_j)}. |
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$$ |
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\end{prop} |
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In other words, if the values of $V$ are clustered, then the corresponding eigenfunctions decay exponentially away from the clusters. The proof immediately follows from the perturbation theory. Note that the statement only makes sense for $\eta\gg 1$, otherwise one can absorb the bound into $C(A,\{A_j\})$. The constant does not depend on $V$ other than through $\eta$. In particular, it does not depend on the energy range inside of a cluster; however, it does depend on the number of lattice sites in each cluster through $C$. |
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\begin{rem} |
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In Proposition \ref{prop_clustering}, one can replace the Laplacian by a weighted Laplacian as long as the weights are bounded from above by $1$, with the same constants. One can also replace it by a long range operator if the distance function is modified accordingly. |
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\end{rem} |
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Finally, we will need the following property related to differentiation of eigenvalues with respect to a parameter. The following is often referred to as Hellmann--Feynman variational formula. |
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\begin{prop} |
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\label{prop_hell} |
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Let $A(t)$ be a family of self-adjoint matrices that depends on a parameter $t$ in a smooth way. Let $\lambda(t)$ be a branch of an isolated eigenvalue of $A(t)$, and let $\psi(t)$ be an $\ell^2$-normalized eigenvector. Then |
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\bee |
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\label{eq_hell} |
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\lambda'(t)=\frac{d}{dt}\<A(t)\psi(t),\psi(t)\>=\<A'(t)\psi(t),\psi(t)\>. |
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\ene |
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\end{prop} |
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\begin{proof} |
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The remaining term in the right hand side of \eqref{eq_hell} is $\lambda(t)\frac{d}{dt}\langle \psi(t),\psi(t)\rangle $, which is identically zero due to normalization of $\psi(t)$. |
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\end{proof} |
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In particular, if $A(t)=A_0+f(t)\<\be_j,\cdot\>\be_j$, then |
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$$ |
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\lambda'(t)=f'(t)\langle\be_j\psi(t),\be_j\rangle=f'(t)|\psi(t;j)|^2. |
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$$ |
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\section{Steps of partial diagonalization and proof of the main result: the one-dimensional case} |
\section{Steps of partial diagonalization and proof of the main result: the one-dimensional case} |
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\begin{document}
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\title[Perturbation theory] {Perturbation theory} \title[Convergence of perturbation theory] {Anderson localization for Maryland-type quasiperiodic operators with a single flat segment} \author[I. Kachkovskiy] {Ilya Kachkovskiy} \address{Department of Mathematics\\ Michigan State University\\ Wells Hall, 619 Red Cedar Rd\\ East Lansing, MI\\ 48824\\ USA} \email{ikachkov@msu.edu} \author[S. Krymskii] {Stanislacv Krymskii} \address{TBC} \email{TBC} \author[L. Parnovski] {Leonid Parnovski} \address{Department of Mathematics\\ University College London\\ Gower Street\\ London\\ WC1E 6BT\\ UK} \email{leonid@math.ucl.ac.uk} \author[R. Shterenberg] {Roman Shterenberg} \address{Department of Mathematics\\ University of Alabama, Birminghan\\ Campbell Hall\\1300 University Blvd\\ Birmingham, AL\\ 35294\\USA } \email{shterenb@math.uab.edu}
%\author[L. Parnovski]
%{Leonid Parnovski}
%\address{Department of Mathematics\\ University College London\\
%Gower Street\\ London\\ WC1E 6BT\\ UK}
%\email{Leonid@math.ucl.ac.uk}
%\keywords{Periodic operators} %\subjclass[2000]{Primary 35P20, 47G30, 47A55; Secondary 81Q10} %\thanks{This work is supported by the Royal Society} %\copyrightinfo{2002}{Alexander V. Sobolev}
\date{\today}
%\begin{abstract} %\input polyabstract.tex %\end{abstract} %\begin{abstract} %We consider a periodic pseudodifferential operator $H=(-\Delta)^l+B$ ($l>0$) in $\R^d$ which satisfies the %following conditions: (i) the symbol of $H$ is smooth in $x$, and (ii) the perturbation %$B$ has order smaller than $2l$. Under these assumptions, we prove that the spectrum %of $H$ contains a half-line.
%\noindent {{\bf AMS Subject Classification:} 35P20 (58J40, 58J50, 35J10)}
%\nl
%{{\bf Keywords:} Bethe-Sommerfeld conjecture, periodic problems, pseudo-differential operators, %spectral gaps} %\end{abstract} \begin{abstract} We prove something\end{abstract} \maketitle \section{Introduction} The present paper can be considered as the direct continuation of the earlier publication \cite{KPS}. We consider quasiperiodic Schr\"odinger operators on $\ell^2(\Z^d)$: \bee \label{eq_h_def} (H(x)\psi)_\bn=\varepsilon(\Delta\psi)_{\bn}+f(x+\bn\cdot\omega)\psi_{\bn}, \ene where $\Delta$ is the discrete Laplacian: $$ (\Delta\psi)_{\bn}=\sum_{\bm\colon |\bm-\bn|_1=1}\psi_\bm, $$ and the function $f$, which generates the quasiperiodic potential, is a non-decreasing function \bee \label{eq_f_infinity} f\colon(-1/2,1/2)\to (-\infty,+\infty),\quad f(-1/2\pm 0)=\mp \infty, \ene and is extended into $\Z\setminus(1/2+\omega\cdot\Z^d)$ by $1$-periodicity. In order for the operator to be well defined, we assume that $x\in \R\setminus(\Z+1/2+\omega\cdot\Z^d)$. We will call potentials of the form \eqref{eq_f_infinity} {\it Maryland-type}, after the classical Maryland model with $f(x)=\tan(\pi x)$. As usual for quasiperiodic operators, the numbers $\{1,\omega_1,\ldots,\omega_d\}$ are assumed to be linearly independent over $\mathbb Q$.
In the present paper, we consider several new classes of $f$ which are not required to be strictly monotone. A model scenario was considered in \cite[Section 6]{KPS}, where monotonicity can be achieved by a simple partial diagonalization of an ``isolated resonance. Here, we consider several more elaborate situations where the resonance is not isolated, but has a one-dimensional propagation character as one changes $x$.
Since we cover several different classes of functions, it is natural to postpone the exact description to the later sections. Here, however, we would like to mention the simplest possible class of operators that we cannot cover by our approach. Suppose that $f$ is constant on an interval $[a,a+L]\subset (-1/2,1/2)$, and suppose that the set $$ S=\{\bn\in\Z^d\colon x+\omega\cdot\bn\in [a,a+L]+\Z\} $$ has an unbounded connected component (here we define connectedness in the natural sense on $\Z^d$). Our methods cannot cover such operators. In fact, we conjecture that such models do not demonstrate Anserson localization around the energy $E=f(h_1)=f(h_2)$ and cannot eliminate the possibility of wave propagation on $S$.
\section{Preliminaries: regularity of $f$ and convergence of perturbation series}
While the functions $f$ under consideration will have flat pieces, a necessary assumption for all proofs below would be sufficiently many pieces with good control of monotonicity. The corresponding regularity conditions and convergence results are summarized in this section.
\subsection{$C_{\reg}$-regularity} Similarly to \cite{KPS}, we will always assume the following:
\begin{itemize}
\item[(f1)] $f\colon (-1/2,1/2)\to \R$ is continuous, $\quad f(-1/2+0)=-\infty,\quad f(1/2-0)=+\infty$, and is extended by $1$-periodicity into $\mathbb R\setminus(\Z+1/2)$.
\end{itemize}
Suppose, $f$ satisfies $\mathrm{(f1)}$. Let
$$
C_{\mathrm{reg}}>0,\quad x_0\in (-1/2,1/2)
$$
We say that $f$ is $C_{\mathrm{reg}}$-regular at $x_0$, if
\begin{itemize}
\item[(cr0)]The pre-image $f^{-1}((f(x_0)-2,f(x_0)+2))\cap (-1/2,1/2)$ is an open interval (denoted by $(a,b)$), and $\left.f\right|_{(a,b)}$ is a one-to-one map between $(a,b)$ and $(f(x_0)-2,f(x_0)+2)$.
\item[(cr1)]Let $D_{\min}(x_0)=\inf\limits_{x\in (a,b)}f'(x)$ (for points where $f'$ does not exist, consider the smallest of the derivative numbers). Then,
\bee
\label{eq_cr1}
D_{\min}(x_0)\le f'(x)\le C_{\mathrm{reg}} D_{\min}(x_0),\quad \forall x\in (a,b).
\ene
\item[(cr2)]Define $(a_1,b_1)=f^{-1}(f(x_0)-1,f(x_0)+1)\subset (a,b)$, and
$$ g(x)=\frac{1}{f(x)-f(x_0)}, \quad x\in (b_1,a_1+1), $$ extended by continuity to $g(\pm 1/2)=0$ (recall that we also assume $f(x+1)=f(x)$, so that the interval $(b_1,a_1+1)$ is essentially $(-1/2,1/2) \setminus(a_1,b_1)$ together with the point $1/2=-1/2\,\,\mathrm{mod}\,\,1$). Then, under the same conventions on the existence of derivatives, $$ |g'(x)|\le C_{\mathrm{reg}} D_{\min}(x_0),\quad x\in (b_1,a_1+1). $$ \end{itemize} \subsection{The frequency vector} As usual for quasiperiodic operators, we will assume $\{1,\omega_1,\ldots,\omega_d\}$ to be linearly independent over $\mathbb Q$. The frequency vector $\omega\in \R^d$ is called {\it Diophantine} if there exist $C_{\dio},\tau_{\dio}$ such that \bee \label{eq_diophantine_definition} \|\bn\cdot\omega\|:=\dist(\bn\cdot\omega,\Z)\ge C_{\dio}|\bn|^{-\tau_{\dio}},\quad \forall \bn\in \Z^d\setminus\{\bze\}. \ene Without loss of generality, we will assume $0<\omega_1<\ldots<\omega_d<1/2$. \subsection{Operators with convergent perturbation series}
In \cite{KPS}, it was shown that, if $f$ is $C_{\reg}$-regular on $(-1/2,1/2)$, if $D_{\min}\ge 1$, and $\omega$ is Diophantine, then the perturbation series converges for sufficiently small $\varepsilon>0$. However, one can also apply the construction from \cite{KPS} in the case where $C_{\reg}$ and $D_{\min}$ themselves depend on $\ep$, however, one needs to be more careful. In \cite[Section 6]{KPS}, an example of such operator was considered. In this section, we will describe a slightly general class of operators for which the construction from the end of \cite[Section 6]{KPS} can be applied, virtually, without any changes. The main results of the present paper will be obtained by reducing various operators to the class described in this section.
We will consider long range quasiperiodic operators with variable hopping terms. A {\it quasiperiodic hopping matrix} is, by definition, a matrix with elements of the following form \bee \label{eq_quasi_1} \Phi_{\bm\bn}(x)=\varphi_{\bm-\bn}(x+(\bm+\bn)\cdot\omega/2),\quad \bm,\bn\in \Z^d, \ene where $\varphi_{\bm}\colon \mathbb R\to \mathbb C$ are Lipschitz $1$-periodic functions, satisfying the self-adjointness condition: $$ \varphi_{\bm}=\overline{\varphi_{-\bm}}. $$ Let also $$ \|\varphi\|_{\infty}=\sup_{\bk} \|\varphi_{\bk}\|_{\infty},\quad\|\varphi'\|_{\infty}=\sup_{\bk} \|\varphi'_{\bk}\|_{\infty}. $$ Define $\range(\Phi)$ to be the smallest number $L\ge 0$ such that $\Phi_{\bn\bm}\equiv 0$ for $|\bm-\bn|>L$. We will only consider hopping matrices of finite range. Note that \eqref{eq_quasi_1} can be reformulated as the following covariance property: $$ \Phi_{\bm+\ba,\bn+\ba}(x)=\Phi_{\bm\bn}(x+\ba\cdot\omega),\quad \bm,\bn,\ba\in \Z^d. $$ Fix some $R\in \mathbb N$, and suppose that $\Phi^1,\Phi^2,\ldots$ is a family of quasiperiodic hopping matrices with $\range(\Phi_k)\le k R$, defined by a family of functions $\varphi^1_{\bm}, \varphi^2_{\bm}, \ldots$. A more general class of operators we would like to consider will be of the following form: \bee \label{eq_longrange_def} H=V+\varepsilon\Phi^1+\varepsilon^2\Phi^2+\ldots,\quad 0\le \varepsilon<1, \ene where $$ (V(x)\psi)_{\bn}=v_{\bn}(x)\psi_{\bn}=f(x_0+\bn\cdot\omega)\psi_{\bn}. $$ One can easily check that, assuming $$ \|\varphi\|_{\infty}=\sup_{j}\|\varphi^j\|_{\infty}=\sup_{j,\bm}\|\varphi^j_{\bm}\|_{\infty}<+\infty,\quad 0\le\varepsilon<1, $$ the part $\Phi=\varepsilon\Phi^1+\varepsilon^2\Phi^2+\ldots$ defines a bounded operator on $\ell^2(\Z^d)$.
The central object of \cite{KPS} is the {\it Rayleigh--Schr\"odinger perturbation series}, which is a formal series of eigenvalues and eigenvectors $$ E=E_0+\varepsilon E_1+\varepsilon^2 E_2+\ldots, $$ $$ \psi=\psi_0+\varepsilon\psi_1+\varepsilon^2\psi_2+\ldots $$ where, in a small departure from the notation of \cite{KPS}, we assume \bee \label{eq_rs_conditions} \lambda_0=f(x_0+\bn\cdot\omega),\quad \psi_0=e_{\bn},\quad \psi_j\perp\psi_0\,\text{ for }j\neq 0. \ene Under the above assumptions, we consider the eigenvalue equation $$ H(x_0)\psi=E\psi $$ as equality of coefficients of two power series: $$ (V+\varepsilon\Phi^1+\varepsilon^2\Phi^2+\ldots)(\psi_0+\varepsilon\psi_1+\varepsilon^2\psi_2+\ldots)= (E_0+\varepsilon E_1+\varepsilon^2 E_2+\ldots)(\psi_0+\varepsilon\psi_1+\varepsilon^2\psi_2+\ldots). $$ Assuming that $f$ is strictly monotone, the above system of equations has a unique solution satisfying \eqref{eq_rs_conditions}.
It will also be convenient to consider a graph $\Gamma(x_0)$, whose set of vertices is $\Z^d$, and there is an edge between $\bm$ and $\bn$ if $\Phi_{\bm\bn}(x_0)\neq 0$. The {\it length} of that edge is the smallest $j>0$ such that $\Phi_{\bm\bn}^j(x_0)\neq 0$.
We can now describe the class of operators. Let $I_1,\ldots,I_k\subset (0,1)$ be disjoint closed intervals, and $\mu_1,\ldots,\mu_k>0$. We will only consider $\mu_j\in \N$ in applications, but the argument also works for $\mu_j\in [0,+\infty)$. We will make the following assumptions. \begin{itemize} \item[(conv1)] $f$ satisfies (f1) from the previous section. Additionally, $f$ is $C_{\reg}$-regular on $(0,1)\setminus (I_1\cup\ldots\cup I_k)$ with $D_{\min}\ge 1$. \item[(conv2)] $c_1 \varepsilon^{\mu_j}\le f'(x)\le c_2$ on $I_j$. \item[(conv3)] Suppose that $x_0+\omega\cdot\bn\in I_j$. For any edge of $\Gamma(x_0)$ that starts at $\bn$, its weight must be strictly greater than $2\mu_j$. \item[(conv4)] The assumptions on the derivative of $\varphi$ can be relaxed as follows: the constants in the estimates will depend, ultimately, on $\max\{\|\varphi\|_{\infty},\varepsilon\|\varphi'\|_{\infty}\}$. In other words, we are allowed for the derivative to be $\varepsilon^{-1}$-large as $\varepsilon\to 0$. The reason is that differentiating $\varphi$ is ``cheaper than differentiating a small denominator, and the latter has already been accounted for in (1). \end{itemize} \begin{prop} \label{prop_convergence} Under the above assumptions $(\mathrm{conv1})$ -- $(\mathrm{conv4})$, there exists $$\ep_0=\ep_0(C_{\reg},\{\mu_j\}_{j=1}^k,C_{\dio},\tau_{\dio},\|\varphi\|_{\infty},\ep\|\varphi'\|_{\infty},c_1,c_2)>0 $$ such that, for $0<\ep<\ep_0$, the perturbation series for the operator \eqref{eq_longrange_def} is convergent, and the operator $H(x)$ satisfies Anderson localization for $x\in \R\setminus(\Z+\omega\cdot\Z+1/2)$. \end{prop}
\section{Some properties of Schr\"odinger eigenvectors}
In this section, we will summarize some basic properties of the discrete Schr\"odinger eigenfunctions in bounded domains. These properties are either well known or straightforward. Let $C\subset \Z^d$. Denote the Laplace operator on $\ell^2(C)$ by $$ (\Delta_C\psi)(\bn)=\sum\limits_{\bm\in C\colon |\bm-\bn|_1=1}\psi(\bm). $$ Let $A,B\subset \Z^d$. We will define $\bn\in A\cup B$ to be {\it simply reachable from $B$} using the following recurrent definition: \begin{enumerate} \item Any point $\bn\in B$ is simply reachable. \item Let $\bn\in A\cup B$, $\bm\in A$, $|\bm-\bn|=1$ and the set $$ \{\bm'\in A\cup B \colon |\bn-\bm'|\le 1,\,\bm'\neq \bm\} $$ is not empty and only consists of simply reachable points. Then we declare $\bm$ to be simply reachable. \end{enumerate} The above definition can be understood as follows. The eigenvalue equation $$ \sum\limits_{\bm\in A\cup B\colon |\bm-\bn|_1=1}\psi(\bm)=E\psi(\bn) $$ relates values of $\psi$ at different points of $A\cup B$. If $\psi(\bn)$ and all $\psi(\bm)$, except for one value $\bm=\bm_0$, are known, the equation lets us determine the remaining value $\psi(\bm_0)$. A simply reachable point, in other words, is a point $\bm\in A\cup B$ such that $\psi(\bm)$ can be determined from the values of $\psi$ on $B$ using a finite sequence of such operations.
We will say that $B$ satisfies {\it simple unique continuation property} (SUP) for $A$ if all points of $A$ are simply reachable from $B$. \begin{lem} \label{lemma_unique} Suppose that $B$ satisfies SUP for $A$. There exists $\delta_0=\delta_0(A,B)>0$ independent of $V$ such that the following is true: for any $M>0$ there is $\delta_1=\delta_1(A,B,M)>0$, also independent of $V$, such that for any $\psi$ satisfying \bee \label{eq_unique_condition} H_{A\cup B}\psi=E\psi,\quad \|\psi\|_{\ell^2(A\cup B)}=1;\quad |\psi(\bm)|<\delta_0\,\,\forall \bm\in B, \ene we have, for some $\bm\in A\cup B$, the following: $$ |(V_\bm-E)\psi(\bm)|\ge \delta_1,\quad |V_\bm-E|\ge M. $$ \end{lem} \begin{rem} For any fixed $V$ and large $M$, one cannot satisfy \eqref{eq_unique_condition}: if a function is small on $B$, then the continuation procedure will result in small values, with bounds that only depend on $B$. Therefore, $|V_{\bm}-E|$ has to be large for some $\bm$. Moreover, since the new values of $\psi$ are calculated using $|V_{\bm}-E|\psi(\bm)|$ rather than $|V_{\bm}-E|$, the former also cannot be too small.
Note that the constants can be chosen independent of $C$, since there are only finitely many possibilities (depending on $A$, $B$) that can possibly affect the results. \end{rem} \begin{proof} The following argument can be easily made more rigorous. Let us start from $A\setminus B$ and gradually apply the eigenvalue equation to recover values of $\psi_{\bm}$ on $B$. At each step, we are obtaining a new value $\psi_{\bn}$ as a combination $\psi_{\bm}$ and $(V_\bm-E)\psi_{\bm}$ already obtained. The initial data is assumed to be small. However, we ultimately obtain an $\ell^2$-normalized eigenvector, and therefore, at some point, we obtain a value $\approx 1$. Consider the first time this happened. The only possibility for that is through $(V_\bm-E)\psi_{\bm}$, where $\psi_{\bm}$ was small, which precisely means that $|V_{\bm}-E|$ must be large. \end{proof} The following version, obtained by simple rescaling, will be important: \begin{cor} In the above result, replace $\Delta$ by $\varepsilon\Delta$ with $\varepsilon>0$. Then the conclusion can be restated as follows: we have, for some $\bm\in A$, $$ |(V_\bm-E)\psi(\bm)|\ge \delta_1 \varepsilon,\quad |V_\bm-E|\ge M\varepsilon. $$ \end{cor} \begin{rem} In Lemma \ref{lemma_unique}, one can replace $\Delta_A$ by a weighted Laplacian with non-zero weights, with the constants depending on upper and lower bounds on weights. Clearly, using zero weights would violate unique continuation. However, the change of weights of the edges connecting vertices in $A\setminus B$ would not affect the result at all, even if we use zero weights, since, once the eigenfunction is fixed in $A\setminus B$, we are only use non-modified edges to perform unique continuation. \end{rem} \begin{rem} \label{remark_extended_sup} In Lemma \ref{lemma_unique}, one can replace $H_A$ by $H_{A'}$, where $A\subset A'$. Suppose that $B\subset A$ satisfies SUP for $H_{A'}$; in other words, any eigenfunction of $H_{A'}$ admits a unique continuation from $A\setminus B$ into $B$ (but not necessarily into $A'$). Then the statement still holds, and does not depend on the form of $H_{A'}$ outside $A$. The proof repeats without changes. \end{rem} We will also need some elementary information about eigenfunction decay, which follows from elementary perturbation theory. \begin{prop} \label{prop_clustering} Let $A$ and $H_A$ as above, and $A=A_1\cup\ldots \cup A_n$ be a disjoint union. Assume that $$ |V_{\bm}-V_{\bn}|\ge\eta,\quad \bm\in A_j,\,\,\bn\in A_j\,\,j\neq k. $$ Then there exists an orthonormal basis $\{\psi_{\bm}\colon \bm\in A\}$ of eigenfunctions of $H_A$ such that if $\bm\in A_j$, then $$ |\psi_{\bm}(\bn)|\le C(A,\{A_j\})\eta_j^{-\dist(\bn,A_j)}. $$ \end{prop} In other words, if the values of $V$ are clustered, then the corresponding eigenfunctions decay exponentially away from the clusters. The proof immediately follows from the perturbation theory. Note that the statement only makes sense for $\eta\gg 1$, otherwise one can absorb the bound into $C(A,\{A_j\})$. The constant does not depend on $V$ other than through $\eta$. In particular, it does not depend on the energy range inside of a cluster; however, it does depend on the number of lattice sites in each cluster through $C$.
\begin{rem} In Proposition \ref{prop_clustering}, one can replace the Laplacian by a weighted Laplacian as long as the weights are bounded from above by $1$, with the same constants. One can also replace it by a long range operator if the distance function is modified accordingly. \end{rem} Finally, we will need the following property related to differentiation of eigenvalues with respect to a parameter. The following is often referred to as Hellmann--Feynman variational formula. \begin{prop} \label{prop_hell} Let $A(t)$ be a family of self-adjoint matrices that depends on a parameter $t$ in a smooth way. Let $\lambda(t)$ be a branch of an isolated eigenvalue of $A(t)$, and let $\psi(t)$ be an $\ell^2$-normalized eigenvector. Then \bee \label{eq_hell} \lambda'(t)=\frac{d}{dt}\<A(t)\psi(t),\psi(t)\>=\<A'(t)\psi(t),\psi(t)\>. \ene \end{prop} \begin{proof} The remaining term in the right hand side of \eqref{eq_hell} is $\lambda(t)\frac{d}{dt}\langle \psi(t),\psi(t)\rangle $, which is identically zero due to normalization of $\psi(t)$. \end{proof} In particular, if $A(t)=A_0+f(t)\<\be_j,\cdot\>\be_j$, then $$ \lambda'(t)=f'(t)\langle\be_j\psi(t),\be_j\rangle=f'(t)|\psi(t;j)|^2. $$
In this section, we will describe several classes of functions $f$ and frequency vectors $\omega$ for which our approach will establish Anderson localization for the operator \eqref{eq_h_def}.
\subsection{Long intervals in 1D} Let $d=1$, and $I=[a,a+L]\subset (-1/2,1/2)$ be an interval. Without loss of generality, assume $0<\omega<1/2$. Additionally, we will require the following:
\begin{enumerate}
\item $L$ is not an integer multiple of $\omega$.
\item $\omega$ is Diophantine.
\item $f(x)=0$ on $[a,a+L]$. Moreover, $f$ is strictly monotone on $(-1/2,a)$ and $(a+L,1/2)$, and is $C_{\reg}$-regular at any $x\in (-1/2,1/2)$ with $|f(x)|\ge 2$.
\item For a constant $C_{\mathrm{sep}}>0$ specified later, we have $C_{\mathrm{sep}}\omega<1-L$.
\end{enumerate}
Note that this implies that $f$ is Lipschitz monotone outside the interval $I$ with a strictly positive lower bound on the derivative. Therefore, $f$ is locally Lipschitz on $(-1/2,1/2)$, but is not differentiable at the endpoints $a,a+L$. Most of these conditions can be relaxed in a more or less straightforward way; for the reader's convenience, we focus on the simplest non-trivial case and discuss generalizations in the later sections. The last assumption is the most important one: the lattice points $\bn$ such that $x+\omega\cdot \bn\in [a,a+L]+\Z$ form clusters with equal values of the potential, creating resonances. It is important that these resonant clusters have bounded size and are disjoint from one another.
\subsection{Long one-frequency intervals for arbitrary $d$} Without loss of generality, assume $0<\omega_1<\ldots<\omega_d<1/2$. Similarly to the previous case, we would like the resonant clusters to be one-dimensional, bounded, and separated from one another. Assume the following conditions: \begin{enumerate} \item $L$ is not an integer multiple of $\omega_1$. \item $\omega$ is Diophantine. \item $f(x)=0$ on $[a,a+L]$. Moreover, $f$ is strictly monotone on $(-1/2,a)$ and $(a+L,1/2)$, and is $C_{\reg}$-regular at any $x\in (-1/2,1/2)$ with $|f(x)|\ge 2$. \item For a constant $C_{\mathrm{sep}}>0$, the following is true: any integer vector $\bn$ with $|\bn|<C_{\mathrm{sep}}$ which is not an integer multiple of $(1,0,\ldots,0)$, we have $\omega\cdot\bn\notin [0,L]+\Z$. \item For a constant $C_{\mathrm{sep}}>0$ specified later, we have $C_{\mathrm{sep}}\omega_1<1-L$. \end{enumerate} Property 4 can be explained in the following way: if an integer vector $\bn$ travels along $\Z^d$ and $x+\omega\cdot\bn\in [L,L+a]+\Z$, then the only way to remain in the interval $[L,L+a]$ is to move along the direction corresponding to $\omega_1$. \subsection{Non-isolated short intervals}
\section{Other} Let $f$ be a function satisfying (f1). Suppose that $\{I_j\}_{j=1}^{N_{\mathrm{int}}}$ is a collection of closed intervals $I_j\subset (-1/2,1/2)$ satisfying the following: \begin{enumerate} \item[(i1)] $f(x)=\mathrm{const}=E_j$ for $x\in I_j$, and all $E_j$ are distinct. By rescaling, we can (and will) without loss of generality assume $|E_i-E_j|>2$ for $i\neq j$. % \item[(i2)] Fix an integer constant $C_{\mathrm{sep}}>0$. For any $x\in I_j$, $|\bn|\le C_{\mathrm{sep}}$ such that $|f(x+\omega\cdot\bn)-E_j|\ge 1$, we will assume that $f$ is $C_{\reg}$-regular at $x+\omega\cdot\bn$. Additionally, if the last condition is replaced by $0<|f(x+\omega\cdot\bn)-E_j|<1$, we assume $1\le f'(x+\omega\cdot\bn)\le C_{\reg}$. \item[(i2)] For any $x\in (-1/2,1/2)$ such that $|f(x)-E_j|\ge 1$ for all $j=1,\ldots,N_{\mathrm{int}}$, assume that $f$ is $C_{\reg}$-regular at $x$. \item[(i3)] Fix an integer constant $C_{\mathrm{sep}}>0$, and suppose that $|\bn-\bm|=1$, $x+\bn\in I_j+\Z$, $x+\bm\notin I_j+\Z$. Then, any lattice path
For any $\bn\in \Z^d$ with $0<|\bn|\le C_{\mathrm{sep}}$ and $\bn_1=0$, and any $x\in I_j$, we have $x+\omega\cdot\bn\notin I_j+\Z$.
\item[(i4)] The numbers $\{|I_1|,\ldots,|I_{N_{\mathrm{int}}}|,\omega_1,\ldots,\omega_d\}$ are rationally independent.
\end{enumerate}
To each interval, one can associate a ``resonance set $\{\bn\in\Z^d\colon x+\omega\cdot\bn\in I_j+\Z\}$. Property $(i2)$ guarantees that each point of each resonance set is surrounded by points where $f$ has proper monotonicity. Note that one cannot assume $C_{reg}$-regularity at all such points, since one can sometimes be very close to the edge of $I_j$. However, property $(i4)$ combined with additional rescaling guarantees that there is at most one such point. At that point, we only require two-sided Lipschitz monotonicity. Property $(i2)$ also guarantees that singular sets corresponding to different intervals are well separated on $\Z^d$. Finally, the most important property $(i3)$ states that singular sets must be one-dimensional integer line segments.
The most simple way to achieve (i2) would be to require $f$ to be $C_{\reg}$-regular \section{Operators with convergent perturbation series}In \cite{KPS}, it was shown that, if $f$ is $C_{\reg}$-regular on $(-1/2,1/2)$, if $D_{\min}\ge 1$, and $\omega$ is Diophantine, then the perturbation series converges for sufficiently small $\varepsilon>0$. However, one can also apply the construction from \cite{KPS} in the case where $C_{\reg}$ and $D_{\min}$ themselves depend on $\ep$, however, one needs to be more careful. In \cite[Section 6]{KPS}, an example of such operator was considered. In this section, we will describe a slightly general class of operators for which the construction from the end of \cite[Section 6]{KPS} can be applied, virtually, without any changes. The main results of the present paper will be obtained by reducing various operators to the class described in this section.
We will consider long range quasiperiodic operators with variable hopping terms. A {\it quasiperiodic hopping matrix} is, by definition, a matrix with elements of the following form \bee \label{eq_quasi_1} \Phi_{\bm\bn}(x)=\varphi_{\bm-\bn}(x+(\bm+\bn)\cdot\omega/2),\quad \bm,\bn\in \Z^d, \ene where $\varphi_{\bm}\colon \mathbb R\to \mathbb C$ are Lipschitz $1$-periodic functions, satisfying the self-adjointness condition: $$ \varphi_{\bm}=\overline{\varphi_{-\bm}}. $$ Let also $$ \|\varphi\|_{\infty}=\sup_{\bk} \|\varphi_{\bk}\|_{\infty},\quad\|\varphi'\|_{\infty}=\sup_{\bk} \|\varphi'_{\bk}\|_{\infty}. $$ Define $\range(\Phi)$ to be the smallest number $L\ge 0$ such that $\Phi_{\bn\bm}\equiv 0$ for $|\bm-\bn|>L$. We will only consider hopping matrices of finite range. Note that \eqref{eq_quasi_1} can be reformulated as the following covariance property: $$ \Phi_{\bm+\ba,\bn+\ba}(x)=\Phi_{\bm\bn}(x+\ba\cdot\omega),\quad \bm,\bn,\ba\in \Z^d. $$ Fix some $R\in \mathbb N$, and suppose that $\Phi^1,\Phi^2,\ldots$ is a family of quasiperiodic hopping matrices with $\range(\Phi_k)\le k R$, defined by a family of functions $\varphi^1_{\bm}, \varphi^2_{\bm}, \ldots$. A more general class of operators we would like to consider will be of the following form: \bee \label{eq_longrange_def} H=V+\varepsilon\Phi^1+\varepsilon^2\Phi^2+\ldots,\quad 0\le \varepsilon<1, \ene where $$ (V(x)\psi)_{\bn}=v_{\bn}(x)\psi_{\bn}=f(x_0+\bn\cdot\omega)\psi_{\bn}. $$ One can easily check that, assuming $$ \|\varphi\|_{\infty}=\sup_{j}\|\varphi^j\|_{\infty}=\sup_{j,\bm}\|\varphi^j_{\bm}\|_{\infty}<+\infty,\quad 0\le\varepsilon<1, $$ the part $\Phi=\varepsilon\Phi^1+\varepsilon^2\Phi^2+\ldots$ defines a bounded operator on $\ell^2(\Z^d)$.
It will also be convenient to consider a graph $\Gamma(x_0)$, whose set of vertices is $\Z^d$, and there is an edge between $\bm$ and $\bn$ if $\Phi_{\bm\bn}(x_0)\neq 0$. The {\it length} of that edge is the smallest $j>0$ such that $\Phi_{\bm\bn}^j(x_0)\neq 0$.
We can now describe the class of operators. Let $I_1,\ldots,I_k\subset (0,1)$ be disjoint closed intervals, and $\mu_1,\ldots,\mu_k>0$. We will only consider $\mu_j\in \N$ in applications, but the argument also works for $\mu_j\in [0,+\infty)$. \begin{enumerate} \item $f$ satisfies (f1) from the previous section. Additionally, $f$ is $C_{\reg}$-regular on $(0,1)\setminus (I_1\cup\ldots\cup I_k)$ with $D_{\min}\ge 1$. \item $f'(x)\ge c_1 \varepsilon^{\mu_j}$ on $I_j$. \item Suppose that $x_0+\omega\cdot\bn\in I_j$. For any edge of $\Gamma(x_0)$ that starts at $\bn$, its weight must be strictly greater than $2\mu_j$. \item The assumptions on the derivative of $\varphi$ can be relaxed as follows: the constants in the estimates will depend, ultimately, on $\max\{\|\varphi\|_{\infty},\varepsilon\|\varphi'\|_{\infty}\}$. In other words, we are allowed for the derivative to be $\varepsilon^{-1}$-large as $\varepsilon\to 0$. The reason is that differentiating $\varphi$ is ``cheaper than differentiating a small denominator, and the latter has already been accounted for in (1). \end{enumerate} \begin{prop} \label{prop_convergence} Under the above assumptions, there exists $$\ep_0=\ep_0(C_{\reg},\{\mu_j\}_{j=1}^k,C_{\dio},\tau_{\dio},\|\varphi\|_{\infty},\ep\|\varphi'\|_{\infty})>0 $$ such that, for $0<\ep<\ep_0$, the perturbation series for the operator \eqref{eq_longrange_def} is convergent and it satisfies Anderson localization for $x\in \R\setminus(\Z+\omega\cdot\Z+1/2)$. \end{prop}
\section{Steps of partial diagonalization and proof of the main result: the one-dimensional case} \subsection{Preliminaries on diagonalization of matrices} Let $A=A(x)=F(x)+G(x)$ be an $N\times N$ matrix-valued function with the following properties. \begin{itemize} \item[(d1)] $F(x)=\diag\{f_0(x),\ldots,f_{n-1}(x)\}$, where $f_j$ are $1$-periodic real-valued functions on $\R$. Each $f_i$ is continuous on $(d_i,d_i+1)$, $f_i(d_i+0)=-\infty$, $f_i(d_i+1-0)=+\infty$. As a consequence, the domain of $f_i$ is $\R\setminus \Z+d_i$. \item[(d2)] $d_0,\ldots,d_{N-1}$ are distinct. Let $\delta=\min_{i\neq j}|d_i-d_j|>0$. \item[(d3)] $G$ is a continuous $1$-periodic function on $\R$, whose values are self-adjoint matrices with real entries. Let $\|G\|_{\infty}=\max_x \|G(x)\|$. \item[(d4)] There is $\delta_1>0$ such that, for all $x$ for which $A(x)$ is defined, the spectrum of $A(x)$ is simple and the eigenvalues are $\delta_1$-separated. \item[(d5)] There is $\delta_2>0$ and a continuous family of functions $f_j(x,t)$, $t\in [0,1]$, such that $f_j(x,0)=f_j(x)$ and $f_j(x,t)=f_j(x)$ whenever $|x-d_j|<\delta_2$. Moreover, the eigenvalues of $A(x,t)$ are uniformly $\delta_1$-separated for all $x,t$, and the eigenvalues of $F(x,1)$ are uniformly $10\|G\|_{\infty}$-separated for all $x$. \end{itemize} \begin{thm} \label{th_diagonalization1} Under the above assumptions, there is a continuous $1$-periodic family of orthogonal matrices $V\colon \R\to O(N)$ such that $V(x)^{-1}A(x)V(x)$ are diagonal, and $j$th column of $V(x)$ approaches the standard basis vector $\be_j$ as $x\to d_j\pm 0$. \end{thm} \begin{proof} Most of the argument follows from analytic perturbation theory. Suppose that $x$ is close to $d_j+0$. Then the matrix $A(x)$ has a large eigenvalue ``attached to $j$th diagonal entry of $F$. Assuming that the corresponding eigenvector has real components and is normalized, there are only two eigenvector branches, approaching $\pm e_j$ as $x\to d_j+0$. Let us choose the ``+ branch. One can check that it has a unique (real normalized) extension into $(d_j,d_j+1)$. In particular, that branch is continuous as $x$ passes the remaining discontinuity points. As $x\to d_j+1-0$, that branch approaches either $\be_j$ or $-\be_j$. It remains to show that the former is actually the case. One can check that the result of such continuation is continuous in the variable $t$ under the assumptions of Property (d5), since the only obstruction to that would be violation of simplicity of the spectrum. Hence, one can check that this is the case for $t=1$. However, if the eigenvalues of the diagonal part are separated, one can check by standard analytic perturbation theory that there are branches of real eigenvectors that are uniformly close to $+\be_j$. \end{proof} We will also need an expanded version of the above statement. Let $A_j(x)=F_j(x)+G_j(x)$, for $j=1,2$, and assume \bee \label{a_def} A(x)=\begin{pmatrix} A_1(x)& B(x)\\ B(x)^T& A_2(x) \end{pmatrix} \ene Suppose that, in the family \eqref{a_def}, the matrix functions $A$, $A_1$, $A_2$ satisfy the assumptions of Theorem \ref{th_diagonalization1}, and, for some $x_0$, we have \bee \label{eq_b_norm} \|B(x_0)\|\le \dist(\sigma(A_1(x_0),\sigma(A_2(x_0)). \ene Let $V(x_0)$ denote the diagonalization of $A(x_0)$ constructed in Theorem \ref{th_diagonalization1}. One can easily see that there are unique real diagonalizations $V_1(x_0)$, $V_2(x_0)$ such that $V(x_0)$ is close to $V_1(x_0)\oplus V_2(x_0)$. We will need to answer the following question: would $V_1(x_0)$, $V_2(x_0)$ coincide with those constructed in Theorem \ref{th_diagonalization1} for $A_1(x)$, $A_2(x)$? \begin{thm} \label{th_diagonalization2} Under the above assumptions, suppose additionally that the family from Property $(d5)$ can be chosen so that \eqref{eq_b_norm} at $x_0$ is satisfied uniformly in $t$. Then the above diagonalizations coincide. \end{thm} The proof can be obtained along the same lines as Theorem \ref{th_diagonalization1}.
\begin{prop} \label{prop_jacobi_separation} Let $J$ be an $n\times n$ matrix satisfying the following properties: $J_{ij}=0$ for $|i-j|>1$, and $J_{ij}=J_{ji}\in \R$. Assume that, $J_{i,i+1}\ge 1$ for all applicable values of $i$. Then the spectrum of $J$ is simple. Moreover, if $E_i\neq E_j$ are two eigenvalues of $J$, then $|E_i-E_j|\ge c(N)>0$, where $c(N)$ is a constant that depends only on $N$. \end{prop} \begin{remark} Since there are no assumptions on the diagonal entries, by rescaling one can obtain a version of the proposition with $|E_i-E_j|\ge c(N)\varepsilon$ provided that $|J_{i,i+1}|>\varepsilon>0$. \end{remark} \subsection{The one-dimensional case} Let $b=a+L/2$ be the center of the flat segment. Let $M=\lceil L/2\omega \rceil+2$, and $H_{M}(x)$ be the restriction of $H(x)$ into the interval $[-M,\ldots,M]$, a $(2M+1)\times (2M+1)$-matrix.
We will construct a conjugation of the family $H(x)$ in several steps. We will start from applying Theorem \ref{th_diagonalization1} to a single block, which will move as $x$ changes. Then, we will introduce multiple shifted copies of the process, ultimately resulting in a covariant diagonalizing operator.
For $x\in [a,a+\omega]$, let $V_0(x)$ be the result of applying Theorem \ref{th_diagonalization1} to the block $H_M(x)$, extended by zeros outside of the block. As a consequence, $V_0(a+\omega)$ will diagonalize the block of $H(\omega)$ corresponding to $[-M,M]$. Let also $T e_j=e_{j+1}$ be the translation operator. Then we have $$ H(a+\omega)=T^{-1}H(a)T. $$ As a consequence, the matrix $T^{-1}V_0(a)T$ diagonalizes the block of $H(a+\omega)$ corresponding to the interval $[-M-1,M-1]$ and, in fact, coincides with the result of Theorem \ref{th_diagonalization1} applied to that block.
We would like to construct an interpolation between $T^{-1}V_0(a)T$ and $V_0(a+\omega)$. Note that, due to Theorem \ref{th_diagonalization2}, they are close to each other, since each of them is close to the result of applying Theorem \ref{th_diagonalization1} to the block corresponding to the interval $[-M-1,M]$. A direct linear interpolation can be easily normalized to be unitary, however, it is not completely obvious how such interpolation would handle off-diagonal elements of $H(x)$. Instead, we will not modify the results of applying Theorem \ref{th_diagonalization1} and will modify $H(x)$ instead. Define $H'(x)$ to be a copy of $H(x)$ with the following modifications: \begin{itemize} \item The coupling between $M-1$ and $M$ will linearly go from $\varepsilon$ to $0$ as $x$ changes from $a$ to $a+\omega$, and is periodically extended into $\R$. \item The coupling between $-M$ and $-M-1$ will linearly go from $0$ to $\varepsilon$ as $x$ changes from $a$ to $a+\omega$, and is periodically extended into $\R$. \end{itemize} Now, apply Theorem \ref{th_diagonalization1} to the block corresponding to the interval $[-M-1,M]$. Note that, at $x=a$ and $x=a+\omega$, the operator further splits into a $1\times 1$ and $(2M+1)\times(2M+1)$ block; as a consequence, the diagonalizing matrix for the interval $[-M-1,M]$ at $x=a$ is the direct sum of the diagonalizing matrix for $[-M,M]$ and the identity; likewise, for $x=a+\omega$ we have the diagonalizing matrix for $[-M-1,M-1]$ augmented by the identity. As a consequence, we obtain the interpolation between $T^{-1}V_0(a)T$ and $V_0(a)$. Denote $$ H_1(x):=V_0(x)^{-1}H(x)V_0(x). $$ \subsection{Properties of $H_1(x)$} At $x=a$, the diagonal matrix elements of $H(a)$ at $[-M+2,M-2]$ correspond to the values of $f$ calculated on the flat segment. The diagonal entries at $-M$, $-M+1$, $M-1$, and $M$ are away from the energy on the flat segment, since they are located on the monotone pieces of $f$. As $x$ increases from $a$ to $a+\omega$, the position of the flat segment relative to the lattice shifts by $1$. The non-trivial part of $V_0(x)$ follows the location of these values.
Recall that $V_0(x)$ is obtained from the matrix diagonalizing the $[-M-1,M]$-block of $H'(x)$. It is easy to see that the diagonal entries of $H_1(x)$ corresponding to $[-M+1,M-2]$, as $a\le x\le a+\omega$, are exactly equal to the corresponding (in the same order) eigenvalues of the $[-M-1,M]$-block of $H'(x)$. All other entries of the $[-M+1,M-2]$-block of $H_1(x)$ are zeros. There are further non-zero entries right outside of that block (away from the diagonal) which will be discussed later. The remaining diagonal entries will be small perturbations of the original entries of $H(x)$ and will preserve the original monotonicity properties.
The central part of the argument is the diagonal entries of $H_1(x)$ at the original location of the flat segment. As discussed above, the study of them reduces to the study of the eigenvalues of $H'_{[-M-1,M]}(x)$ as $a\le x\le a+\omega$. The diagonal entries at $-M-1,-M,-M+1,M-2,M-1,M$ are separated from the rest of the spectrum and generate four ``perturbative eigenvalues, for which there is a convergent power series expansion in $\varepsilon$ whose coefficients are Lipschitz functions of $x$ (with the first term being the original diagonal entry). These eigenvalues will be called regular. Among the remaining eigenvalues, the block $[-M+3,M-4]$ is always on the flat segment, and the corresponding eigenvalues will be called singular. Among the remaining two, $-M+2$ and $M-3$, at least one is separated from the flat segment by an amount independent of $\varepsilon$, and the other may or may not be on (or close to) the flat segment. We will call these two eigenvalues transitional.
Assume that $x\in [a,a+\omega]$ and let us summarize some properties of $H_1(x)$. \begin{enumerate} \item $H_1(x)$ differs from $H(x)$ only inside of the block corresponding to $[-M-2,M+1]$. The entries on the boundary of the above block are small Lipschitz continuous perturbations of the original entries, and do not affect the rest of the argument. \item The perturbative eigenvalues of $H_1(x)$ inside the above block are small Lipschitz perturbations of the original eigenvalues and, again, do not affect the rest of the argument (one can relax all Lipschitz constants by a factor of $2$ and assume that $\varepsilon$ is small). \item Proposition \ref{prop_clustering} imply the following magnitude bounds on the columns of $V_0(x)$. If $\psi$ is an eigenvector corresponding to a resonant or a transitional eigenvalue, we have $$ \psi(-M-1)=O(\varepsilon^3),\,\psi(-M)=O(\varepsilon^2),\,\psi(-M+1)=O(\ep); $$ $$ \psi(M)=O(\varepsilon^3),\,\psi(M-1)=O(\varepsilon^2),\,\psi(M-2)=O(\ep). $$ The perturbative eigenvectors are supported at $-M-1$, $-M$, $-M+1$, $M-2$, $M-1$, $M$, respectively, and decay exponentially away from the support point (unlike the resonant or possibly transitional eigenvectors who are spread around $[-M+2,M-3]$). At least one of the transitional eigenvectors, actually, behaves as a perturbative eigenvector supported at $-M+2$ or $M-3$, depending on $x$. The decay is potentially better for some values of $x$, since the Laplacian in $H'$ has actually been partially cut off. Below is an illustration with 4 resonant eigenvalues, two transitional, and three perturbative eigenvalues on each side, a total of 12: $$ V_0(x)\approx\begin{pmatrix} 1&\ep&\ep^2&\ep^3&\ep^3&\ep^3&\ep^3&\ep^3&\ep^3&\ep^9&\ep^{10}&\ep^{11}\\ \ep&1& \ep& \ep^2 & \ep^2& \ep^2& \ep^2& \ep^2& \ep^2& \ep^8&\ep^{9}&\ep^{10}\\ \ep^{2}&\ep& 1& \ep & \ep& \ep& \ep& \ep& \ep& \ep^7& \ep^8&\ep^{9}\\ \ep^{3}&\ep^2&\ep& 1& 1& 1& 1& 1& 1& \ep^6&\ep^7&\ep^{8}\\ \ep^{4}&\ep^3&\ep^2& 1& 1& 1& 1& 1& 1& \ep^5&\ep^6&\ep^{7}\\ \ep^{5}&\ep^4&\ep^3& 1& 1& 1& 1& 1& 1& \ep^4&\ep^5&\ep^{6}\\ \ep^{6}&\ep^5&\ep^4& 1& 1& 1& 1& 1& 1& \ep^3&\ep^4&\ep^{5}\\ \ep^{7}&\ep^6&\ep^5& 1& 1& 1& 1& 1& 1& \ep^2&\ep^3&\ep^{4}\\ \ep^{8}&\ep^7&\ep^6& 1& 1& 1& 1& 1& 1& \ep&\ep^2&\ep^{3}\\ \ep^{9}&\ep^8& \ep^7& \ep & \ep& \ep& \ep& \ep& \ep& 1& \ep&\ep^2\\ \ep^{10}&\ep^9 & \ep^8& \ep^2 & \ep^2& \ep^2& \ep^2& \ep^2& \ep^2& \ep& 1&\ep\\ \ep^{11}&\ep^{10}&\ep^9&\ep^3&\ep^3&\ep^3&\ep^3&\ep^3&\ep^3&\ep^2&\ep&1 \end{pmatrix} $$ Moreover, the result of applying Proposition \ref{prop_clustering} is differentiable in $x$. Therefore, matrix elements of $V_0(x)$ are Lipschitz continuous with the Lipschitz that is $\ep^{-1}$ worse than the $L^{\infty}$ bound.
Moreover, perturbative eigenvalues are actually Lipschitz continuous with only an $O(1)$ loss, since they are separated from the rest of the spectrum.
\item Let $E$ be the value of $f$ on the flat segment. Each singular eigenvalue $\lambda(x)$ satisfies $|\lambda-E|\le C_1 \varepsilon$ where $C_1$, essentially, only depends on the size of the flat segment. Apply Lemma \ref{lemma_unique} to a singular eigenvalue with $M\gg C_1$. We can see that the corresponding eigenfunction $\psi(x)$ has an entry $|\psi(x;\bm)|\ge C_2\varepsilon$, and $f(x+\bm\cdot\omega)$ cannot be equal to $E$. Here, $C_2$ also does not depend on $x$ or $\varepsilon$. As a consequence, Proposition \ref{prop_hell} implies that $\lambda'(x)\ge C_3 \varepsilon^2$, where $C_3$ does not depend on $\varepsilon$ and is can be chosen uniformly in $x$.
Alternatively, one can use Remark \ref{remark_extended_sup} and the fact that each (two-point) interval $[-M+1,-M+2]$ and $[M-3,M-2]$ satisfy SUP for the lattice points corresponding to singular eigenvalues. Among the above intervals, one can pick the one whose transitional eigenvalue is perturbative and apply the argument to that eigenvalue, without considering cases.
\item A similar argument as above also applies to transitional eigenvalues, however, in this case one has to consider two regimes. If the transition eigenvalue is close to $E$, say, by $100 C_1\varepsilon$, then one can choose $M\gg 1000 C_1$ and apply the above construction. If the transitional eigenvalue is away from $E$, by at least $100 C_1 \varepsilon$, then it, essentially, becomes perturbative, and admits a complete expansion in the powers of $1/100$. Assuming that the unperturbed eigenvalue is Lipschitz monotone with constant 1, we we can still claim that its Lipschitz monotonicity was not violated. In general, we may need to replace $1/100$ by a quantity that depends on $C_{\reg}$, but, in the end of the day, it will reduce to freedom in choosing $M$ in Lemma \ref{lemma_unique}. \item In order to determine all changes in off-diagonal entries of $H$ after conjugation by $V_0(x)$, one needs to extend the size by 1 on each side, and calculate the following, where $V_0(x)$ denotes the $12\times 12$ block $$ \begin{pmatrix} 1&0&0\\ 0&V_0(x)^T&0\\ 0&0&1 \end{pmatrix}\cdot \begin{pmatrix} O(1)&\ep&0&0&0&0&0\\ \ep&0&\ep&0&0&0&0\\ 0&\ep&0&0_{1\times 8}&0&0&0\\ 0&0&0_{8\times 1}&0_{8\times 8}&0_{8\times 1}&0&0\\ 0&0&0&0_{1\times 8}&0&\ep&0\\ 0&0&0&0&\ep&0&\ep\\ 0&0&0&0&0&\ep&O(1)\\ \end{pmatrix}\cdot \begin{pmatrix} 1&0&0\\ 0&V_0(x)&0\\ 0&0&1 \end{pmatrix} $$ Note that the extra blocks $\begin{pmatrix}0&\ep\\\ep&0\end{pmatrix}$ on each side appeared because $V_0(x)$ was diagonalizing a modified Hamiltonian, with changed components of the Laplacian. After conjugation, all off-diagonal entries connecting resonant or transitional eigenvalues with likewise, will vanish. Off-diagonal entries connecting perturbative eigenvalues with perturbative, ultimately, are $O(\ep)$ with Lipschitz bound $O(1)$, or better, and therefore do not affect the perturbative expansion. All we need to consider is entries connecting resonant/transitional eigenvalues with perturbative eigenvalues. Each such entry is obtained \item Let us now focus on the off-diagonal elements of $H_1(x)$. A convenient way of looking at it would be to subtract from $H(x)$ the block $[-M+1,M-2]$, which is completely diagonalized by $V_0(x)$. Thus, we need to consider $$ H_2(x)=V_0(x)^{-1}(H(x)-H_{[-M+1,M+2](x)})V_0(x). $$ The convenience of the above construction is that, whenever we are multiplying $V_0$ and $H(x)-H_{[-M+1,M+2](x)}$, their supports almost do not overlap, and in the end we only need to consider the boundary effects. \end{enumerate} \section{The general higher-dimensional construction} Consider the operator \eqref{eq_h_def} on $\ell^2(\Z^d)$ and assume that $f$ satisfies (f1). Assume, without loss of generality, that $$ 0<\omega_1<\omega_2<\ldots<\omega_d<1/2. $$ Fix some $C_{\reg}>0$ and let $$ I_{\sing}=\{x_0\in(-1/2,1/2)\colon f\text{ is not }C_{\reg}\text{-regular at }x_0\}. $$ Introduce also the set of singular lattice points: $$ S_{\sing}(x_0)=\{\bn\in \Z^d\colon x_0+\bn\in I_{\sing}+\Z\}. $$ Any lattice point that is not singular will be called regular. The results \cite{KPS} imply that, if $\bn$ is regular, then $H(x_0)$ has an eigenfunction which is a small perturbation of $\be_{\bn}$ and is given by a convergent perturbation series started at that point.
Let $S\subset \Z^d$ be a finite subset potentially containing some singular points. Let also $R\supset S$ be another finite subset, with the following properties: \begin{enumerate} \item $S\cup(S+\be_{1})\subset R\cap (R+\be_1)$. \item $(R\cap (R+\be_1))\setminus(S\cup (S+\be_{1}))$ satisfies SUP for $S\cup (S+\be_{1})$. \item For $x\in [x_0,x_0+\omega_1]$, all points of $R\setminus(S\cup (S+\be_{1}))$ are regular. \end{enumerate} Let also $R_-=R\setminus(R+\be_1)$, $R_+=(R+\be_1)\setminus R$, $R_0=R\cap(R+\be_1)$, $R'=R\cup(R+\be_1)$ We can represent $R$ as a disjoint union $R=R_-\cup R_+\cup R_0$. Denote by $H'_{R'}(x)$ the following family of operators on $\ell^2(R')$: $$ H'_{R'}(x_0)=H_{R_-\cup R_0}(x_0)\oplus H_{R_+}(x_0); $$ $$ H'_{R'}(x_0+\omega)=H_{R_-}(x_0+\omega)\oplus H_{R_0\cup R_+}(x_0+\omega), $$ with linear interpolation of the off-diagonal terms in between; one can also formally write it as $$ H'_{R'}(x)=(1-t)\left(H_{R_-\cup R_0}(x)\oplus H_{R_+}(x)\right)+t\left(H_{R_-}(x)\oplus H_{R_0\cup R_+}(x_0+t\omega_1)\right),\quad x=x_0+t\omega_1, $$ for $t\in [0,1]$. Assume the following condition: \begin{itemize} \item[(gen1)] The spectra of $H'_{R'}(x)$ are simple for all $x\in [x_0,x_0+\omega_1]$. \end{itemize} Denote by $V_{R'}(x)$, $x\in [x_0,x_0+\omega_1]$, the result of applying Theorem \ref{th_diagonalization1} to the family $H'_{R'(x)}$. These operators act in $\ell^2(R')$. Extend them into $\ell^2(\Z^d)$ by identity, and denote the extension by $V(x)$. Similarly to the one-dimensional case, we have $$ V(x_0+\omega_1)=T^{-1} V(x) T, $$ where $(Tf)(\bn)=f(\bn+\be_1)$. Using the last formula, one can extend $V(x)$ into $R$.
Now, denote $$ H'(x)=V(x)^{-1}H(x)V(x). $$ The above construction will be called {\it diagonalization of a moving block}. Note that the singular set has the following covariance property: $$ S_{\sing}(x_0)=S_{\sing}(x_0-\omega\cdot\bn)+\bn. $$ As a consequence, if $S$ from the moving block construction contains singular points for $H(x_0)$, they will naturally correspond to singular points in $S-\be_1$ for $H(x_0+\omega_1)$. In the above construction, as $x$ increases, the set $S$ `moves' along $\Z^d$ with speed $1/\omega_1$ and traces a part of $S_{\sing}$, thus justifying the name of the construction.
Suppose that we were able to cover $S_{\sing}$ by moving blocks such that the corresponding sets $R'$ do not overlap (in other words, all operators $V'(x)$ corresponding to different blocks commute). Assume also that $R'$ are large enough. Then, there is a possibility to reduce the final operator $H'(x)$ to the case described in Proposition \ref{prop_convergence}. Indeed, suppose that, for a single block, we have $R'$ large enough, and there is a subset $R\subset R'$ that still satisfies SUP for $S\cup(S+\be_1)$ within $R'$. Then: \begin{enumerate} \item The eigenvectors of $H'(x)$ can be split into regular (perturbative, supported outside of $S\cup (S+\be_1)$) and singular (supported in $S\cup (S+\be_1)$). For each regular eigenvector, the corresponding eigenvalue is a small perturbation of $f(x_0+\omega\cdot \bn)$ and therefore is Lipschitz monotone. For each singular eigenvalue, the corresponding eigenvector has small but non-trivial part supported on $R$. Therefore, it will inherit Lipschitz monotonicity of the corresponding regular eigenvalue from $R$, with weakening by a power of $\varepsilon$, depending only on the geometry of $R$. Thus, assumption (2) before Proposition \ref{prop_convergence} will be satisfied. \item The diagonalization will eliminate all off-diagonal entries connecting $S\cup (S+\be_1)$ with entries from $R_0$. If $R_0$ is large enough, this means that all off-diagonal entries coming out of $S\cup (S+\be_1)$ will be of the order at least $\dist(\Z^d\setminus R_0,S\cup (S+\be_1)$. Therefore, if $R_0$ is large enough, one can satisfy the assumption (3) before Proposition \ref{prop_convergence}, independently of the geometry of $R$. \end{enumerate} %\subsection{The higher-dimensional case with multiple low frequencies} % However, this methods makes the second step of diagonalization much more complicated, because one has to remove the nondiagonal elements. As a result, if $|\omega_j\pm k\omega_1|< 2h$ (where $k$ can be any number such that $k\omega_1\leq 2h$), one will end up with extra connections between the vertices where $|\sum \omega_in_i - a|\leq h,$ which have weight of the order $O(\epsilon)$ and cannot be destroyed. Therefore, successful diagonalization requires that these connections were unable to form an infinite system at the same $x$. But they can form such a system iff the connections with $\omega_i$ have formed such a system. Therefore, if there is no infinite system, then the diagonalization is possible. \begin{thebibliography}{99} \bibitem{Arnold} Arnol'd~V., {\it Remarks on perturbation theory for problems of Mathieu type} (Russian), Uspekhi Mat. Nauk 38 (1983), no. 4 (232), 189 -- 203. \bibitem{Bellissard} Béllissard~J., Lima~R., Scoppola~E., {\it Localization in $\nu$-dimensional incommensurate structures}, Comm. Math. Phys. 88 (1983), no. 4, 465 -- 477. \bibitem{Eliasson}Eliasson~H., {\it Absolutely convergent series expansions for quasi periodic motions}, Math. Phys. Electron. J. 2 (1996), Paper 4, 33 pp. \bibitem{FP}Figotin~A., Pastur~L., {\it An exactly solvable model of a multidimensional incommensurate structure}, Comm. Math. Phys. 95 (1984), no. 4, 401 -- 425. \bibitem{GFP} Grempel~D., Fishman~S., Prange~R., {\it Localization in an incommensurate potential: an exactly solvable model}, Phys. Rev. Lett. 49 (1982), 833 -- 836. \bibitem{Simon_maryland}Simon~B., {\it Almost periodic Schr\"odinger operators. IV. The Maryland model}, Ann. Phys. 159 (1985), no. 1, 157 -- 183. \bibitem{Wencai} Jitomirskaya~S., Liu~W., {\it Arithmetic spectral transitions for the Maryland model}, Comm. Pure Appl. Math. 70 (2017), no. 6, 1025 -- 1051. \bibitem{JK}Jitomirskaya~S., Kachkovskiy~I., {\it All couplings localization for quasiperiodic operators with Lipschitz monotone potentials}, J. Eur. Math. Soc. 21 (2019), no. 3, 777 -- 795. \bibitem{JY2}Jitomirskaya~S., Yang~F., {\it Pure point spectrum for the Maryland model: a constructive proof}, Ergodic Theory Dynam. Systems, in press (2020). \bibitem{JY}Jitomirskaya~S., Yang~F., {\it Singular continuous spectrum for singular potentials}, Comm. Math. Phys. 351 (2017), no. 3, 1127 -- 1135. \bibitem{Ilya}Kachkovskiy~I., {\it Localization for quasiperiodic operators with unbounded monotone potentials}, J. Funct. Anal 277 (2019), no. 10, 3467 -- 3490. \bibitem{KPS}Kachkovskiy~I., Parnovski~L., Shterenberg~R., {Convergence of perturbation series for unbounded monotone quasiperiodic operators} \end{thebibliography}\end{document}