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Ben
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My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the eigenvalues of the $n \times n$ matrix, $H_n = \Delta_n + V_n$ where $\Delta_n$ is the discrete Laplacian on $\mathbb{Z}_n$ and $V_n$ is a diagonal matrix whose entries are iid, mean zero with finite variance.

Let $\Lambda_n$ be the the eigenvalues of $H_n$. They lie in $(-2,2)$ and so the spacings (at least not near the edge of the spectrum) are on the order of $1/n$. And so one can zoom in on the process near an $E \in (-2,2)$ and consider the point process $$P_n(E) = n(\Lambda_n - E).$$

What I would like to know is if there are any estimates on the quantity $|P_N(E) \cap K|$, where $K$ is a compact subset of $\mathbb{R}$. This is the number of eigenvalues in a microscopic interval around $E$. In particular is it true that that there is a $p>1$ and some constants $C(n,E)$ depending on $n$ and $E$ such that $$ \mathbb{E} \left[ |P_n(E) \cap K|^p \right] \leq C(n,E). $$ I'm hoping that $\sup_{E \in (-2,2)} \sup_n C(n,E) < \infty$$\sup_n \sup_{E \in (-2,2)} C(n,E) < \infty$.

Thanks in advance for any help or sources you can point me to.

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the eigenvalues of the $n \times n$ matrix, $H_n = \Delta_n + V_n$ where $\Delta_n$ is the discrete Laplacian on $\mathbb{Z}_n$ and $V_n$ is a diagonal matrix whose entries are iid, mean zero with finite variance.

Let $\Lambda_n$ be the the eigenvalues of $H_n$. They lie in $(-2,2)$ and so the spacings (at least not near the edge of the spectrum) are on the order of $1/n$. And so one can zoom in on the process near an $E \in (-2,2)$ and consider the point process $$P_n(E) = n(\Lambda_n - E).$$

What I would like to know is if there are any estimates on the quantity $|P_N(E) \cap K|$, where $K$ is a compact subset of $\mathbb{R}$. This is the number of eigenvalues in a microscopic interval around $E$. In particular is it true that that there is a $p>1$ and some constants $C(n,E)$ depending on $n$ and $E$ such that $$ \mathbb{E} \left[ |P_n(E) \cap K|^p \right] \leq C(n,E). $$ I'm hoping that $\sup_{E \in (-2,2)} \sup_n C(n,E) < \infty$.

Thanks in advance for any help or sources you can point me to.

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the eigenvalues of the $n \times n$ matrix, $H_n = \Delta_n + V_n$ where $\Delta_n$ is the discrete Laplacian on $\mathbb{Z}_n$ and $V_n$ is a diagonal matrix whose entries are iid, mean zero with finite variance.

Let $\Lambda_n$ be the the eigenvalues of $H_n$. They lie in $(-2,2)$ and so the spacings (at least not near the edge of the spectrum) are on the order of $1/n$. And so one can zoom in on the process near an $E \in (-2,2)$ and consider the point process $$P_n(E) = n(\Lambda_n - E).$$

What I would like to know is if there are any estimates on the quantity $|P_N(E) \cap K|$, where $K$ is a compact subset of $\mathbb{R}$. This is the number of eigenvalues in a microscopic interval around $E$. In particular is it true that that there is a $p>1$ and some constants $C(n,E)$ depending on $n$ and $E$ such that $$ \mathbb{E} \left[ |P_n(E) \cap K|^p \right] \leq C(n,E). $$ I'm hoping that $\sup_n \sup_{E \in (-2,2)} C(n,E) < \infty$.

Thanks in advance for any help or sources you can point me to.

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Ben
Ben
  • 195
  • 1
  • 6

Finite Volume 1D Anderson Tight Binding Model

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the eigenvalues of the $n \times n$ matrix, $H_n = \Delta_n + V_n$ where $\Delta_n$ is the discrete Laplacian on $\mathbb{Z}_n$ and $V_n$ is a diagonal matrix whose entries are iid, mean zero with finite variance.

Let $\Lambda_n$ be the the eigenvalues of $H_n$. They lie in $(-2,2)$ and so the spacings (at least not near the edge of the spectrum) are on the order of $1/n$. And so one can zoom in on the process near an $E \in (-2,2)$ and consider the point process $$P_n(E) = n(\Lambda_n - E).$$

What I would like to know is if there are any estimates on the quantity $|P_N(E) \cap K|$, where $K$ is a compact subset of $\mathbb{R}$. This is the number of eigenvalues in a microscopic interval around $E$. In particular is it true that that there is a $p>1$ and some constants $C(n,E)$ depending on $n$ and $E$ such that $$ \mathbb{E} \left[ |P_n(E) \cap K|^p \right] \leq C(n,E). $$ I'm hoping that $\sup_{E \in (-2,2)} \sup_n C(n,E) < \infty$.

Thanks in advance for any help or sources you can point me to.