Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in $d\le 3$

Abstract

We consider the continuous Anderson Hamiltonian with white noise potential on $(-L/2,L/2)^d$ in dimension $d\le 3$, and derive the asymptotic of the smallest eigenvalues when L goes to infinity. We show that these eigenvalues go to $-\infty$ at speed $(\log L)^{1/(2-d/2)}$ and identify the prefactor in terms of the optimal constant of the Gagliardo–Nirenberg inequality. This result was already known in dimensions 1 and 2, but appears to be new in dimension 3. We present some conjectures on the fluctuations of the eigenvalues and on the asymptotic shape of the corresponding eigenfunctions near their localisation centers.

Appendices

Appendix A: Renormalisation constants

We let G be the Green’s function of $-\Delta$, and $P^{(m)}$ be the Green’s function of $-\Delta +m$, we refer to [18, Sec 3.1] for the expressions. Recall that $n_m$ is the smallest integer such that $2^{-n_m} \le 1/\sqrt{m}$.

Lemma A. 1

Fix $r>0$. For every $m\ge 1$, there exists a decomposition $P^{(m)} = P^{(m)}_+ + P^{(m)}_-$ such that:

• 1. $P^{(m)}_+$ is supported in $B(0,2^{-n_m+1})$ and satisfies $P^{(m)}_+ = P^{(m)}$ on $B(0,2^{-n_m-1})$, while $P^{(m)}_-$ is $\mathcal {C}^\infty$ and vanishes on $B(0,2^{-n_m-1})$.

• 2. For all $k\in \mathbf {N}^d$ such that $|k|< r$ we have $\int x^k P^{(m)}_+(x) dx = 0$.

• 3. There exists a constant $C>0$, independent of $m\ge 1$, such that for all $k\in \mathbf {N}^d$ such that $|k|< r$ we have

  $$\begin{aligned} |\partial ^k P^{(m)}_+(x)| \le C | \partial ^k G(x)|,\quad x\in \mathbf {R}^d. \end{aligned}$$

Proof

This is a consequence of [18, Lemma 3.1] except for the property “$P^{(m)}_+ = P^{(m)}$ on $B(0,2^{-n_m-1})$” that was not stated there. However, this property follows if one picks carefully the functions $\eta _k$ in that proof: namely, it suffices to impose to the functions $\eta _k$ to be supported in $B(0,1) \backslash B(0,1/2)$. This can always be achieved, see for instance [5, Lemma 8.1]. $\square$

We introduce the renormalisation constants as follows. In dimension 1, we set $C_\varepsilon ^{(m)}(\beta ) := 0$. In dimension 2, we set

$$\begin{aligned} C_\varepsilon ^{(m)}(\beta ) := \beta ^2 \int _{\mathbf {R}^2} P_+^{(m)}(x) \rho _\varepsilon ^{*2}(x) dx. \end{aligned}$$

A computation shows that there exists a constant $\tilde{c}_\rho (m)$ independent of $\beta$ such that $C_\varepsilon ^{(m)}(\beta ) = \beta ^2 (2\pi )^{-1} \ln \varepsilon ^{-1} + \beta ^2 \tilde{c}_\rho (m) + o(1)$ as $\varepsilon \downarrow 0$.

In dimension 3, we set $C_\varepsilon ^{(m)}(\beta ) := \beta ^2 c_\varepsilon ^{(m)} + \beta ^4 c_\varepsilon ^{(m),1,1} + \beta ^4 c_\varepsilon ^{(m),1,2}$ where

$$\begin{aligned} c_\epsilon ^{(m)}&:= \int P_+^{(m)}(x) \rho _\varepsilon ^{*2}(x) \,dx,\\ c^{(m),1,1}_\epsilon&:= \iiint P_+^{(m)}(x_1) P_+^{(m)}(x_2) P_+^{(m)}(x_3) \rho _\varepsilon ^{*2}(x_1+x_2)\rho _\varepsilon ^{*2}(x_2+x_3) \,dx_1 \,dx_2 \,dx_3,\\ c^{(m),1,2}_\epsilon&:= \iiint P_+^{(m)}(x_1) P_+^{(m)}(x_2) \big (P_+^{(m)}(x_3) \rho _\varepsilon ^{*2}(x_3) - c_\epsilon \delta _0(x_3) \big )\rho _\varepsilon ^{*2}(x_1+x_2+x_3) \\&\quad dx_1 \,dx_2 \,dx_3. \end{aligned}$$

There exist some constants $c_\rho , \tilde{c}_\rho , \tilde{c}_\rho ^{1,1}(m),c_\rho ^{1,2}$ independent of $\beta$ such that as $\varepsilon \downarrow 0$

$$\begin{aligned} c_\epsilon ^{(m)}&= \frac{c_\rho }{\varepsilon } + \tilde{c}_\rho \sqrt{m} + o(1),\\ c^{(m),1,1}_\epsilon&= \ln \frac{1}{\varepsilon } + \tilde{c}_\rho ^{1,1}(m) + o(1),\\ c^{(m),1,2}_\epsilon&= c_\rho ^{1,2} + o(1). \end{aligned}$$

Note that the only constant that depends on m is $\tilde{c}_\rho ^{1,1}(m)$, and its expression is a bit involved so we refrain from writing it explicitly. On the other hand, if we let $G(x) = \frac{1}{4\pi |x|}$ (which is nothing but the Green’s function of $-\Delta$), we have $c_\rho = \int _{\mathbf {R}^3} G(y) \rho ^{*2}(y) dy$, $\tilde{c}_\rho =- \int _{\mathbf {R}^3} G(y)|y| \rho ^{*2}(y) dy$, and

$$\begin{aligned}&c_\rho ^{1,2} = \iiint G(y_1) G(y_3)\left( G(z_2 - y_3) - G(z_2) \right. \\&\quad -\left. \left\langle \nabla G(z_2), y_3 \right\rangle \right) \rho ^{*2}(y_3) \rho ^{*2}(y_1 + z_2) ~dy_1 dz_2 dy_3. \end{aligned}$$

The construction of the renormalised model $Z_\varepsilon ^{(m)}(\beta )$ follows along the lines of [16] and [18]. However, we take slightly different renormalisation constants compared to [18]: instead of taking the constants built from the kernel $P^{(m)}_+$, we take those associated to $P^{(1)}_+$. Namely, in dimension 2, we take $C_\varepsilon := C_\varepsilon ^{(1)}$ and in dimension 3, we take the three constants $c_\varepsilon ^{(1)}$, $c_\varepsilon ^{(1),1,1}$ and $c_\varepsilon ^{(1),1,2}$. This produces a limiting renormalised model that differs from the one in [18] by finite constants, as shown by the above asymptotics: this does not modify the final operator, but greatly simplify its definition (in particular, one does not need to deal with constants like $C^{(m)-(1)}$ as in [18]).

Let us finally mention that the renormalised model satisfies for $d=2$

$$\begin{aligned} \Pi _x^{(m),\varepsilon }(\beta ) \Xi \mathcal {I}(\Xi )(x) = - C_\varepsilon (\beta ), \end{aligned}$$

and for $d=3$

$$\begin{aligned}&\Pi _x^{(m),\varepsilon }(\beta ) \Xi \mathcal {I}(\Xi )(x) = - c_\varepsilon ^{(1)}(\beta ),\quad \Pi _x^{(m),\varepsilon }(\beta ) \Xi \mathcal {I}(\Xi \mathcal {I}(\Xi \mathcal {I}(\Xi )))(x) \\&\quad = - c_\varepsilon ^{(1),1,1}(\beta ) - c_\varepsilon ^{(1),1,2}(\beta ). \end{aligned}$$

Appendix B: Canonical model

The aim of this section is to construct a canonical admissible model $Z^{(m)}(V) := (\Pi ^{(m)}_x(V), \Gamma ^{(m)}_{xy}(V))$ associated to some potential function $V\in L^2_{\tiny \text{ loc }}(\mathbf {R}^d)$ and to show that

$$\begin{aligned} \sup _{m\ge 1} {\left| \!\left| \!\left| Z^{(m)}(V);Z^{(m)}(0) \right| \!\right| \!\right| }_Q < \infty . \end{aligned}$$

(3.5)

for all box $Q \subset \mathbf {R}^d$. If V were smooth, then [14, Prop. 8.27] would ensure that the model is admissible. Moreover, once the model is defined, (3.5) essentially follows from Lemma A. 1 since the kernel $P^{(m)}_+$ is controlled by G uniformly over all m. However, here V is only in $L^2_{\tiny \text{ loc }}(\mathbf {R}^d)$, which necessitates some adjustments in order to obtain the required analytical bounds. In the sequel, we fix $V \in L^2_{\tiny \text{ loc }}(\mathbf {R}^d)$.

The set of symbols $\mathcal {T}= \mathcal {U}\cup \mathcal {F}$ introduced in Sect. 3.1 can be obtained through a recursive construction: let $\mathcal {U}_0 = \{\mathbf {1}, X^k \}$, and for $n\ge 0$ we define recursively

$$\begin{aligned} \mathcal {F}_n&:= \{\Xi \tau : \tau \in \mathcal {U}_n\},\\ \mathcal {U}_{n+1}&:= \{\mathbf {1}, X^k\} \cup \{\mathcal {I}(\tau ): \tau \in \mathcal {F}_{n}, |\tau |+2 < \gamma \}, \end{aligned}$$

for $\gamma = 2 - 4\kappa$. Subsequently we have $\mathcal {F}= \bigcup _{n \ge 0} \mathcal {F}_n$ and $\mathcal {U}= \bigcup _{n\ge 0} \mathcal {U}_n$. Note that all elements in $\mathcal {U}$ are of positive homogeneity.

Given this recursive structure, we can define the model $\Pi ^{(m)}$ in the following manner: for $m \ge 1$, $x,y \in \mathbf {R}^d$, we set

$$\begin{aligned} \Pi ^{(m)}_x \mathbf {1}(y) = 1,\quad \Pi ^{(m)}_x X^k(y) = (y - x)^k,\quad \Pi ^{(m)}_x \Xi (y) = V(y) \end{aligned}$$

and then recursively

$$\begin{aligned} \Pi ^{(m)}_x (\tau \Xi )(y)&= (\Pi ^{(m)}_x \tau )(y) \cdot V(y), \\ \Pi ^{(m)}_x (\mathcal {I}{\bar{\tau }})(y)&= \int P^{(m)}_+(y - z) \Pi ^{(m)}_x {\bar{\tau }}(z) dz \\&\quad - \sum _{|k| < |{\bar{\tau }}| + 2} \frac{(y - x)^k}{k!} \int D^k P^{(m)}_+(x - z) \Pi ^{(m)}_x {\bar{\tau }}(z) dz \end{aligned}$$

for all $\tau \in \mathcal {U}$ and ${\bar{\tau }} \in \mathcal {F}$. Since V is a function, all these expressions are well-defined.

We need the following estimate, that follows from standard arguments based on Lemma A. 1. If $f_x$ is a function that satisfies

$$\begin{aligned} |f_x(y)| \lesssim |x-y|^\zeta , \end{aligned}$$

uniformly over all $y\in \mathbf {R}^d$ such that $|x-y| \le C$, then

$$\begin{aligned}&\left| \int P^{(m)}_+(y - z) V(z) f_x(z) dz - \sum _{|k| < |\tau | + 2} \frac{(y - x)^k}{k!} \int D^k P^{(m)}_+(x - z) V(z)f_x(z) dz\right| \\&\quad \lesssim |x-y|^{\zeta +2-\frac{d}{2}}, \end{aligned}$$

uniformly over all $y\in \mathbf {R}^d$ such that $|x-y|\le C-1$ and over all $m\ge 1$.

Let us now prove recursively the required analytical bounds on $\Pi ^{(m)}$ for some fixed box Q. Pick a large constant $C>0$. Suppose that for all $\tau \in \mathcal {U}_n$, we have $|\Pi ^{(m)}_x \tau (y)| \lesssim |y - x|^{|\tau |}$ uniformly over all $x \in Q$ and all $y\in \mathbf {R}^d$ such that $|x-y|\le C$ and over all $m\ge 1$. For any $\tau \in \mathcal {U}_n$ we have, by the Cauchy-Schwarz inequality

$$\begin{aligned} \big |\langle \Pi _x^{(m)} \Xi \tau ,\varphi _x^\lambda \rangle \big | \lesssim \lambda ^{|\tau |} \int |V(y)| |\varphi _x^\lambda (y)| dy \le \lambda ^{|\tau |} \Vert V\, \mathbf {1}_{B(x,C)} \Vert _{L^2} \lambda ^{-\frac{d}{2}} \lesssim \lambda ^{|\tau \Xi |}, \end{aligned}$$

uniformly over all $x \in Q$, all $m\ge 1$, all $\varphi \in \mathscr {B}^r$ and al $\lambda \in (0,C]$. Furthermore, by the estimate above

$$\begin{aligned} \big |\Pi _x \mathcal {I}(\Xi \tau )(y)\big | \le |x-y|^{|\tau |+2 - \frac{d}{2}}, \end{aligned}$$

uniformly over all $x \in Q$ and all $y\in \mathbf {R}^d$ such that $|x-y|\le C-1$ and over all $m\ge 1$.

Since only finitely many iterations suffice to exhaust the whole set $\mathcal {U}\cup \mathcal {F}$, we deduce that

$$\begin{aligned} \left| \left\langle \Pi ^{(m)}_x \tau , \varphi ^\lambda _x \right\rangle \right| \lesssim \lambda ^{|\tau |} \end{aligned}$$

uniformly over all $x\in Q$, all $m\ge 1$, all $\varphi \in \mathscr {B}^r$, all $\lambda \in (0,1]$ and all $\tau \in \mathcal {U}\cup \mathcal {F}$.

Regarding the construction of $\Gamma ^{(m)}$, we argue that it is uniquely determined once $\Pi ^{(m)}$ is specified on the negative levels of the regularity structure, see for example [17, Thm. 2.10]. Now that the model $Z^{(m)}(V)$ is defined with respect to $V \in L^2_{\tiny \text{ loc }}$ and that the bound on ${\left| \!\left| \!\left| \Pi ^{(m)} \right| \!\right| \!\right| }_Q$ is independent of m, we can invoke [18, Lem. 2.3] to conclude ${\left| \!\left| \!\left| Z^{(m)}(V); Z^{(m)}(0) \right| \!\right| \!\right| }_Q$ is bounded by a constant independent of m, whence (3.5) follows.

Appendix C: Proof of Proposition 2.5

The goal of this subsection is to prove Proposition 2.5, which is basically an extension of a result by Gärtner and König [12] where they have considered the case of smooth bounded potential. We begin by proving a variation to their Proposition 1 in [12].

Proposition C. 1

For fixed $L> r > 0$ and any bounded smooth potential V, there exists a constant $K > 0$ such that

$$\begin{aligned} \lambda (Q_L, V) \ge \min _{k \in \mathbf {Z}^d : |k|_\infty \le \frac{L}{2r} + \frac{3}{4} } \lambda (rk + Q_{3r/2}, V) - \frac{K}{r^2}. \end{aligned}$$

Proof

The proof is built upon a specific choice of partition of unity: Let $\eta : \mathbf {R}^d \rightarrow [0, 1]$ be a smooth function supported in $Q_{3r/2}$ such that it gives 1 on $Q_{r/2}$, $\sum _{k \in \mathbf {Z}^d} \eta _k^2(x) = 1$ and that $\sum _{k \in \mathbf {Z}^d} |\nabla \eta _k|^2(x) \le K/r^2$ for all $x \in \mathbf {R}^d$, where $\eta _k(x) := \eta (r k + x)$. We will give a construction of such partition later in the proof.

Note first that we have the following variational formulation for the principal eigenvalue of the operator $-\Delta + V$ on a domain $D \subset \mathbf {R}^d$:

$$\begin{aligned} \lambda (D, V) = \inf _{\begin{array}{c} \psi \in C^\infty _c(D) \\ \Vert \psi \Vert _{L^2} = 1 \end{array}} \int _{\mathbf {R}^d} |\nabla \psi |^2 + V \psi ^2 =: \inf _{ \Vert \psi \Vert _{L^2} = 1} G^V(\psi ). \end{aligned}$$

Given the desired partition of unity $(\eta _k)$, we take $\psi \in C^\infty _c(Q_L)$ such that $\Vert \psi \Vert _{L^2} = 1$ and set $\psi _k = \eta _k \psi$. With the fact that $|\nabla \psi _k|^2 = \eta _k^2 |\nabla \psi |^2 + \psi ^2 |\nabla \eta _k|^2 + \nabla (\eta _k^2) \cdot \nabla (\psi ^2) /2$, it follows that $\sum _k |\nabla \psi _k|^2 = |\nabla \psi |^2 + \sum _k |\nabla \eta _k|^2 \psi ^2$. Therefore

$$\begin{aligned} \sum _{k \in \mathbf {Z}^d} \Vert \psi _k\Vert _{L^2}^2 G^V(\psi _k/\Vert \psi _k\Vert _{L^2}) = \sum _{k \in \mathbf {Z}^d} \int _{\mathbf {R}^d} \left( |\nabla \psi _k|^2 + V \psi _k^2\right) \le G^{V}(\psi ) + \frac{K}{r^2} \end{aligned}$$

where we have used the property $\sup _{x \in \mathbf {R}^d} \sum _{k} |\nabla \eta _k|^2(x) \le K/r^2$ in the last inequality. Since $\psi$ is supported in $Q_L$, the sum over k is in fact a finite sum as we can restrict ourselves to those k’s such that $r|k|_\infty - 3r/4 < L/2$. Hence

$$\begin{aligned} G^{V} (\psi ) + \frac{K}{r^2}\ge & {} \sum _{|k|_\infty< \frac{L}{2r} + \frac{3}{4}} \Vert \psi _k\Vert _{L^2}^2 \min _{k \in \mathbf {Z}^d : |k|_\infty< \frac{L}{2r} + \frac{3}{4} } \lambda (Q_{kr+3r/2}, V) \\= & {} \min _{|k|_\infty < \frac{L}{2r} + \frac{3}{4} } \lambda (Q_{kr+3r/2}, V). \end{aligned}$$

We then have our desired inequality by taking an infimum over $\psi$.

Finally, we finish this proof by constructing the function $\eta$ with desired properties. The d-dimensional construction can be first reduced to a 1-dimensional one by setting $\eta (x) = \zeta (x_1) \dots \zeta (x_d)$ for $x = (x_1, \dots , x_d) \in \mathbf {R}^d$ with $\zeta$ being the 1-dimensional version of $\eta$. On the other hand $\zeta$ can be constructed as follows. Let $\varphi (x) := c \int _{-\infty }^x e^{-1/(1 - u^2)} \mathbf {1}_{|u| < 1}$, where the constant c is chosen so that $\varphi (x) = 1$ for $x \ge 1$. Note that $\varphi (x) = 0$ for $x\le -1$ and that $\varphi (x)+\varphi (-x) = 1$ for all $x\in \mathbf {R}$. One can also verify that $\sqrt{\varphi }$ is smooth. Now set

$$\begin{aligned} \zeta (x) = \sqrt{\varphi (2(r + 2x)/r) \varphi (2(r - 2x)/r)}. \end{aligned}$$

One can see that $\zeta (x) = 1$ if $|x| \le r/4$, $\zeta (x) = 0$ if $|x| > 3r/4$ and $\sum _k \zeta ^2(rk + x) = 1$ for all x. Moreover, since the function $\varphi$ is independent of r, we have $\Vert \nabla \eta \Vert _{\infty } \lesssim 1/r$ with the proportionality constant depending only on the function $\varphi$, thus giving the bound $\sup _{x \in \mathbf {R}^d} \sum _{k} |\nabla \eta _k|^2(x) \le K/r^2$ for some constant $K > 0$. $\square$

Proof of Proposition 2.5

To prove Proposition 2.5, we first consider the same assertions with $\xi$ replaced by the mollified and renormalized white noise $\xi _\varepsilon$. In this case, the lower bound of (2.1) follows from Proposition C. 1, while the remaining assertions are consequences of the variational formuation of eigenvalues

$$\begin{aligned} \lambda _n(D, V) := \inf _{\begin{array}{c} F \sqsubset C^\infty _c(D)\\ \mathrm {dim}(F) = n \end{array}} \sup _{\begin{array}{c} \psi \in F\\ \left\| \psi \right\| _{L^2} = 1 \end{array}} G^V(\psi ) \end{aligned}$$

(where the functional $G^V$ is defined in the proof of Lemma C. 1) for any domain $D \subset \mathbf {R}^d$ and bounded smooth potential V.

Assertions being established for all smooth bounded V, now it remains to take $V_\varepsilon = \beta \xi _\varepsilon + C_\varepsilon (\beta )$ and to pass to the limit. By Proposition 2.1, the eigenvalues of the renormalized Hamiltonian $\mathcal {H}(Q_L,\beta \xi _{\varepsilon _k} + C_{\varepsilon _k}(\beta ))$ converge almost surely to those of $\mathcal {H}(Q_L,\beta \xi )$, which implies immediately the desired almost sure inequality (2.1). $\square$