Level Statistics for One-Dimensional Schrödinger Operators and Gaussian Beta Ensemble

Abstract

We study the level statistics for two classes of 1-dimensional random Schrödinger operators: (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result (Kotani and Nakano in Festschrift Masatoshi Fukushima, 2013) imply the coincidence of the limits of circular and Gaussian beta ensembles.

Appendix

In this section we recall basic tools used in this paper. The content below are borrowed from [2]. For $f \in C^{\infty }(M)$ let $R_{\beta } f :=(L + i\beta )^{-1}f$ $(\beta > 0)$, $R f :=L^{-1}(f - \langle f \rangle )$. Then by Ito’s formula,

$$\begin{aligned} \int \limits _0^t e^{i \beta s} f(X_s) ds&= \left[ e^{i \beta s} (R_{\beta }f)(X_s) \right] _0^t + \int \limits _0^t e^{i \beta s} d M_s(f, \beta )\\ \int \limits _0^t f(X_s) ds&= \langle f \rangle t + \left[ (R f) (X_s) \right] _0^t + M_t(f,0). \end{aligned}$$

$M_s(f, \beta ), M_s(f, 0)$ are the complex martingales whose variational process satisfy

$$\begin{aligned} \langle M(f, \beta ), M(f, \beta ) \rangle _t&= \int \limits _0^t [ R_{\beta } f, R_{\beta } f] (X_s) ds,\\ \left\langle M(f, \beta ), \overline{M(f, \beta )} \right\rangle _t&= \int \limits _0^t \left[ R_{\beta } f, \overline{R_{\beta } f}\right] (X_s) ds\\ \langle M(f, 0), M(f, 0) \rangle _t&= \int \limits _0^t [ R f, R f] (X_s) ds,\\ \left\langle M(f, 0), \overline{M(f, 0)} \right\rangle _t&= \int \limits _0^t \left[ R f, \overline{R f}\right] (X_s) ds \end{aligned}$$

where

$$\begin{aligned}{}[f_1, f_2](x)&:= L(f_1 f_2)(x) - (Lf_1)(x) f_2(x) - f_1(x) (Lf_2)(x)\\&= (\nabla f_1, \nabla f_2) (x). \end{aligned}$$

Then the integration by parts gives us the following formulas to be used frequently.

Lemma 5.15

$$\begin{aligned} (1)&\int \limits _0^t b(s) e^{i \beta s} e^{i \gamma \tilde{\theta }_s} f(X_s) ds\\&= \left[ b(s) e^{i \gamma \tilde{\theta }_s} e^{i \beta s} (R_{\beta }f)(X_s) \right] _0^t - \int \limits _0^t b'(s) e^{i \gamma \tilde{\theta }_s} e^{i \beta s} (R_{\beta }f)(X_s)ds\\&-\, \frac{i \gamma }{2 \kappa } \int \limits _0^t b(s) a(s) Re (e^{2i \theta _s}-1) e^{i \gamma \tilde{\theta }_s} e^{i \beta s} F(X_s) (R_{\beta }f) (X_s) ds\\&+ \int \limits _0^t b(s) e^{i \beta s} e^{i \gamma \tilde{\theta }_s} d M_s(f, \beta ). \end{aligned}$$

$$\begin{aligned} (2)&\int \limits _0^t b(s) e^{i \gamma \tilde{\theta }_s} f(X_s) ds\\&= \langle f \rangle \int \limits _0^t b(s) e^{i \gamma \tilde{\theta }_s} ds\\&+ \left[ b(s) e^{i \gamma \tilde{\theta }_s} (Rf)(X_s) \right] _0^t - \int \limits _0^t b'(s) e^{i \gamma \tilde{\theta }_s} (Rf)(X_s) ds\\&-\, \frac{i \gamma }{2 \kappa } \int \limits _0^t a(s)b(s) Re (e^{2i \theta _s} - 1) e^{i \gamma \tilde{\theta }_s} F(X_s) (Rf)(X_s) ds\\&+\, \int \limits _0^t b(s) e^{i \gamma \tilde{\theta }_s} dM_s(f, 0). \end{aligned}$$

We will also use following notation for simplicity.

$$\begin{aligned} g_{\kappa }&:= (L+2i \kappa )^{-1}F, \quad g := L^{-1}(F - \langle F \rangle ),\\ h_{\kappa , \beta }&:= (L + 2i \beta \kappa )^{-1}F g_{\kappa }\\ M_s(\kappa )&:= M_s (F, 2 \kappa ), \quad M_s := M_s (F, 0),\\ \widetilde{M}_s^{(\beta )}(\kappa )&:= M_s (F g_{\kappa }, \beta \kappa ), \quad \widetilde{M}_s := M_s (F g_{\kappa }, 0). \end{aligned}$$

Lemma 5.16

Let $\Psi _n, n=1,2, \ldots$, and $\Psi$ are continuous and increasing functions defined on a open set $K \subset \mathbf{R}$ such that $\lim _{n \rightarrow \infty }\Psi _n(x)=\Psi (x)$ pointwise.

If $y_n \in Ran \;\Psi _n$, $y \in Ran \;\Psi$ and $y_n \rightarrow y$, then it holds that

$$\begin{aligned} \Psi _n^{-1}(y_n) \mathop {\rightarrow }\limits ^{n \rightarrow \infty } \Psi ^{-1}(y). \end{aligned}$$