On Derivation of the Poisson–Boltzmann Equation

Abstract

Starting from the microscopic reduced Hartree–Fock equation, we derive the macroscopic linearized Poisson–Boltzmann equation for the electrostatic potential associated with the electron density.

Appendices

Appendix A: $\epsilon (T) \rightarrow \epsilon (0)$ as $T \rightarrow 0$

Lemma A 1

Let $\text {xc}= 0$. Then $\epsilon \equiv \epsilon (T) \rightarrow \epsilon (0)$ as $T \rightarrow 0$, where $\epsilon (0)$ is the dielectric constant for $T=0$ obtained in [7].

Proof

We see from (1.35) below that $\epsilon (T), T=1/\beta ,$ is of the form

$$\begin{aligned} \epsilon (T) = \frac{1}{2\pi i} \int _\Gamma f_T(z-\mu ) X(z) \end{aligned}$$

(A.1)

where X(z) is some holomorphic function on $\mathbb {C}\backslash \mathbb {R}$, independent of $\beta$, and remains holomorphic on the real axis where the gap of $h_{\mathrm{per}}$ occurs. On $\mathbb {R}$, we note that $f_{\mathrm{FD}}(\beta x)$ converges to the indicator function $\chi _{(-\infty , 0)}$ as $\beta \rightarrow \infty$. If we take $\beta \rightarrow \infty$, the integral

$$\begin{aligned} \frac{1}{2\pi i} \int _\Gamma f_T(z-\mu ) X(z) \end{aligned}$$

(A.2)

converges to $\frac{1}{2\pi i} \int _{G_1} X(z)$ where $G_1$ is any contour around the part of the spectrum of $h_{\mathrm{per}}$ that is less than $\mu _{\mathrm{per}}$. This is the same expression as in [7] after inserting $1 = \sum _i |\varphi _i \rangle \langle \varphi _i|$ for each resolvent of $h_{\mathrm{per}}$ in X(z) where the $\varphi _i$’s are eigenvectors of $h_{\mathrm{per}}$. $\square$

Appendix B: Bounds on m and V

In this section, we prove bounds on m and V given (1.33) and (4.28). Note that $m=\Vert V\Vert _{L^1_{\mathrm{per}}}$. Since $f'_{T} < 0$ ($T= 1/\beta$), (4.28) implies that $V > 0$ and therefore, by (4.28), $\Vert V\Vert _{L^1_{\mathrm{per}}} = \int _\Omega V$, where $\Omega$ is a fundamental domain of $\mathcal {L}$ (see Sect. 1.5), which yields

$$\begin{aligned} m=\int _\Omega V = - \mathrm {Tr}_{L^2_{\mathrm{per}}} f'_{T}(h_{\mathrm{per}, 0}-\mu ),\ \qquad \mu =\mu _{\mathrm{per}}. \end{aligned}$$

(B.1)

Lemma B.1

Let Assumption [A1] hold and $\eta _0$ be given in (1.27). Then

$$\begin{aligned} m=\Vert V\Vert _{L^1_{\mathrm{per}}} \ge \frac{1}{4} \beta e^{-\beta \eta _0}, \end{aligned}$$

(B.2)

where $\eta _0$ is given in (1.27).

Proof

Using that $\eta _0$ is the smallest distance between $\mu = \mu _{\mathrm{per}}$ and the spectrum of $h_{\mathrm{per}, 0}$ (see (1.27)) and Eq. (B.1) and replacing $\mathrm {Tr}_{L^2_{\mathrm{per}}} f'_{T}(h_{\mathrm{per}, 0}-\mu )$ by the contribution of the eigenvalue of $h_{\mathrm{per}, 0}$ closest to $\mu$, we find

$$\begin{aligned} m \ge&- f'_{T}(\eta _0) = \beta \frac{e^{\beta \eta _0}}{(1+e^{\beta \eta _0})^2} \ge \frac{1}{4} \beta e^{-\beta \eta _0}. \end{aligned}$$

(B.3)

This gives (B.2). $\square$

Lemma B.2

Let Assumption [A1] hold. Then, for $1 \le p \le \infty$,

$$\begin{aligned} \Vert V\Vert _{L^p_{\mathrm{per}}} \lesssim \beta e^{-\eta _0\beta }. \end{aligned}$$

(B.4)

Proof

We do the case for $p=1$ and $p=\infty$, and conclude the lemma by interpolation. By Assumption [A1], the potential $\phi _{\mathrm{per}}$ is bounded. Thus, $h_{\mathrm{per}, 0}$ has only discrete spectrum on $L^2_{\mathrm{per}}(\mathbb {R}^3)$ and

$$\begin{aligned} \frac{1}{\beta }\Vert V\Vert _{L^1_{\mathrm{per}}}&= \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0})} \frac{ e^{\beta (\lambda -\mu )}}{(1+e^{\beta (\lambda -\mu )})^2} \end{aligned}$$

(B.5)

$$\begin{aligned}&\le \sum _{\mu > \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{\beta (\lambda -\mu )}+ \sum _{\mu < \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{-\beta (\lambda -\mu )} \end{aligned}$$

(B.6)

$$\begin{aligned}&= \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0})}e^{\beta |\lambda -\mu |} . \end{aligned}$$

(B.7)

Again, we use that $\eta _0$ is the smallest distance between $\mu = \mu _{\mathrm{per}}$ and the spectrum of $h_{\mathrm{per}, 0}$ (see (1.27)). Peeling the eigenvalue(s) closest to $\mu$ and letting $\eta _0+\xi$ stand for the distance between $\mu$ and the rest of the spectrum $\sigma (h_{\mathrm{per}, 0})$, we find, for some constant c,

$$\begin{aligned} \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0})}&e^{\beta |\lambda -\mu |} =ce^{\beta \eta _0}+ \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0}), |\lambda -\mu |\ge \eta _0+\xi }e^{\beta |\lambda -\mu |}. \end{aligned}$$

(B.8)

We estimate the sum on the r.h.s. by an integral as follows. Since the potential $\phi _{\mathrm{per}}$ is infinitesimally bounded with respect to $-\Delta$, the eigenvalues of $h_{\mathrm{per}, 0}$ go to infinity at a similar rate as those of $-\Delta$ (on $L^2_{\mathrm{per}}(\mathbb {R}^3)$), i.e. as $n^2$. Thus, assuming that for $\lambda$ sufficiently large, the nth eigenvalue $\lambda _n\approx n^2$ has the degeneracy of the order $O(n^k), k\ge 0$, we conclude that

$$\begin{aligned}&\sum _{\mu < \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{-\beta (\lambda -\mu )}\lesssim \int _{x^2 \ge \mu +\eta _0+\xi } x^k e^{-\beta (x^2 -\mu )} d x \nonumber \\&\quad =\frac{1}{2} \int _{y\ge \eta _0+\xi } (y+\mu )^{\frac{k-1}{2}} e^{-\beta y} d y \lesssim \frac{1}{\beta }\mu ^{\frac{k-1}{2}} e^{-\beta (\eta _0+\xi )}. \end{aligned}$$

(B.9)

For the first sum in (B.6), we consider separately the cases $\mu \lesssim 1$ and $\mu \gg 1$ and, in the 2nd case, break the sum into the sums over $\lambda \lesssim 1$ and $\lambda \gg 1$. In the first three situations, the estimate is straightforward and in the last one, we proceed as in (B.9) to obtain

$$\begin{aligned} \sum _{\mu -\eta _0-\xi \ge \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{\beta (\lambda -\mu )}\lesssim \frac{1}{\beta }\mu ^{\frac{k-1}{2}} e^{-\beta \eta _0}. \end{aligned}$$

This proves the lemma for $p=1$.

Let $W^{4,1}_{\mathrm{per}}$ be the usual Sobolev space associated to $L^1_{\mathrm{per}}$ involving up to 4 derivatives. For the case $p=\infty$, we use the Sobolev inequality

$$\begin{aligned} \Vert f\Vert _\infty \lesssim \Vert f\Vert _{W^{4,1}_{\mathrm{per}}} \end{aligned}$$

(B.10)

for $f \in W^{4,1}_{\mathrm{per}}$. Thus, it suffices for us to estimate $\Vert \nabla ^j V\Vert _{L^1_{\mathrm{per}}}, j=0,\ldots ,4$. To this end, we note that

$$\begin{aligned} \nabla {\text {den}}(A) = {\text {den}}( [\nabla , A] ) \end{aligned}$$

(B.11)

for an operator A on $L^2_{\mathrm{per}}(\mathbb {R}^3)$. Thus, it suffices that we estimate the trace 1-norm of $\nabla ^s f_{T}'(h_{\mathrm{per}, 0}-\mu ) \nabla ^{4-s}$ on $L^2_{\mathrm{per}}(\mathbb {R}^3)$ for $s=0,\ldots ,4$. Since the potential $\phi _{\mathrm{per}}$ is bounded together with all its derivatives, we have, for $s=0, \dots , 4$,

$$\begin{aligned} \Vert \nabla ^s h^{-s/2}\Vert \lesssim 1, \end{aligned}$$

(B.12)

where $h:=h_{\mathrm{per}, 0}+c$, with $c>0$ s.t. $h_{\mathrm{per}, 0}+c>0$. Indeed, to fix ideas, consider one of the terms, say, $\Vert \nabla ^3 h^{-3/2}\Vert$. We have $\Vert \nabla ^3f\Vert ^2\le \Vert (-\Delta )^{3/2}f\Vert ^2=\langle f, (h+\phi _{\mathrm{per}}-c)^3f\rangle$. Taking $f = h^{-3/2}u$, expanding the binomial $(h+\phi _{\mathrm{per}}-c)^3$ and commuting the operator h in the resulting terms $h^2\phi _{\mathrm{per}}$ and $\phi _{\mathrm{per}} h^2$ to the right and left, respectively, and estimating the resulting commutators, $[h, \phi ]$ and $[\phi _{\mathrm{per}}, h]=-[h, \phi _{\mathrm{per}}]$, we arrive at the estimate $\Vert \nabla ^3 h^{-3/2}\Vert \lesssim 1$ as claimed. (B.12) implies also that $\Vert h^{-2+s/2}\nabla ^{4-s}\Vert \lesssim 1$, for $j=0, \dots , 4$. As the result, we have

$$\begin{aligned} \Vert \nabla ^s f_{T}'(h_{\mathrm{per}, 0}-\mu ) \nabla ^{4-s}\Vert _{S^1}\lesssim \Vert g(h_{\mathrm{per}, 0})\Vert _{S^1}, \end{aligned}$$

where $g(x):=-(x+c)^{s/2} f_{T}'(x-\mu ) (x+c)^{2-s/2}\ge 0$. Hence, it suffices to estimate $\Vert g(h_{\mathrm{per}, 0})\Vert _{S^1}=\mathrm {Tr}[g(h_{\mathrm{per}, 0})]$. The latter can be done the same way as the case for $p=1$ by summing eigenvalues of $h_{\mathrm{per}, 0}$ and the lemma is proved. $\square$

Appendix C: Bound on $M_\delta$

In an analogy to $L_{\mathrm{per}}^2\equiv L_{\mathrm{per}}^2(\mathbb {R}^3)$ given in (1.19), we let

$$\begin{aligned} L^2_{\mathrm{per}, \delta }\equiv L^2_{\mathrm{per}, \delta }(\mathbb {R}^3) := \{ f \in L^2_{\mathrm{loc}}(\mathbb {R}^3) : f \text { is } \mathcal {L}_\delta \text {-periodic } \}. \end{aligned}$$

(C. 1)

Moreover, we recall $\nabla ^{-1} := \nabla (-\Delta )^{-1}$ (see (3.56)). The main result of this appendix is the following

Proposition C.1

Let Assumption [A1] hold. Then the operator $M_\delta$ can be decomposed as

$$\begin{aligned} M_\delta = M_\delta ' + M_\delta '' \, , \end{aligned}$$

(C. 2)

with the operator $M_\delta '$ and $M_\delta ''$ satisfying the estimates

$$\begin{aligned}&\Vert {\bar{P_r}} \nabla ^{-1} M_\delta ' P_r\varphi \Vert _{L^2} \lesssim \delta ^{-1} \Vert V\Vert _{L^2_{\mathrm{per}}} \Vert P_r\varphi \Vert _{L^2}, \end{aligned}$$

(C. 3)

$$\begin{aligned}&\Vert {\bar{P_r}} \nabla ^{-1} M_\delta '' P_r\nabla ^{-1} \varphi \Vert _{L^2} \lesssim \Vert P_r\varphi \Vert _{L^2}. \end{aligned}$$

(C. 4)

Proof of Proposition C.1

Proposition 4.2 and the rescaling relation (3.34) imply the explicit form for the k-fibers of $M_\delta$:

Lemma C.2

Then $M_\delta$ has a Bloch–Floquet decomposition (2.34) with $\mathcal {L}= \mathcal {L}_\delta$, whose $k-$fiber $M_{\delta , k}$ acting on $L^2_{\mathrm{per}, \delta }$ is given by

$$\begin{aligned} M_{\delta ,k} f =&- \delta {\text {den}}\left[ \oint r^\delta _{\mathrm{per, 0}}(z)f r^\delta _{\mathrm{per}, k}(z)\right] \end{aligned}$$

(C. 5)

where $f \in L^2_{\mathrm{per}, \delta }$ and, on $L^2_{\mathrm{per}, \delta }$,

$$\begin{aligned}&r^\delta _{\mathrm{per}, k}= (z- h^\delta _{\mathrm{per}, k})^{-1}, \quad h^\delta _{\mathrm{per}, k}= \delta ^2 (-i\nabla -k)^2 + \delta \phi _{\mathrm{per}}^\delta . \end{aligned}$$

(C. 6)

We decompose the operator $M_{\delta , k}$ acting on $L^2_{\mathrm{per}, \delta }$ as

$$\begin{aligned} M_{\delta , k}&=: M_{\delta , 0} + M_{\delta , k}'' \, , \end{aligned}$$

(C. 7)

where $M_{\delta , 0} = M_{\delta , k=0}$ and $M_{\delta , k}'$ is defined by the expression (C. 7). We define operators $M_\delta '$ and $M_\delta ''$ on $L^2(\mathbb {R}^2)$ via

$$\begin{aligned} M_\delta '&:= \int _{\Omega _\delta ^*}^\oplus d{\hat{k}} \, M_{\delta , 0} \varphi , \end{aligned}$$

(C. 8)

$$\begin{aligned} M_\delta ''&:= \int _{\Omega _\delta ^*}^\oplus d{\hat{k}} \, M''_{\delta , k} \varphi \, , \end{aligned}$$

(C. 9)

where $\Omega _\delta ^*$ is a fundamental cell of the reciprocal lattice to $\mathcal {L}_\delta$ and $d{\hat{k}} = |\Omega _\delta ^*|^{-1} dk$. By Lemma C.2 and definition (C. 7), the latter operators satisfy (C. 2).

Lemma C.3

$M_\delta '$ (see (C. 8)) restricted to the range of $P_r$ is a multiplication operator given by

$$\begin{aligned} (M_\delta ' P_r\varphi )(x) = V_\delta (x) (P_r\varphi )(x), \end{aligned}$$

(C. 10)

where

$$\begin{aligned} V_\delta (x) = -\delta ^{-2} {\text {den}}\left[ f_{T}'(h_{\mathrm{per}, 0}-\mu ) \right] (\delta ^{-1}x) \, , \end{aligned}$$

(C. 11)

with $h_{\mathrm{per}, 0}$ given in (4.6) (with $k = 0$).

Proof

By (C. 5) and definition of $M_\delta '$ in (C. 8), we see that

$$\begin{aligned} M_\delta '&P_r \varphi = - \int ^\oplus _{\Omega _\delta ^*} d{\hat{k}} \, \delta {\text {den}}\left[ \oint r^\delta _{\mathrm{per, 0}}(z)(P_r \varphi )_k r^\delta _{\mathrm{per, 0}}(z)\right] . \end{aligned}$$

(C. 12)

By Corollary 2.4 and the Cauchy integral formula,

$$\begin{aligned} M_\delta ' P_r\varphi&= - \int ^\oplus _{\Omega _\delta ^*} d{\hat{k}} \, \delta {\text {den}}\left[ \oint (r^\delta _{\mathrm{per, 0}}(z))^2 \right] |\Omega _\delta |^{-1} {\hat{\varphi }}(k) \end{aligned}$$

(C. 13)

$$\begin{aligned}&= - \int ^\oplus _{\Omega _\delta ^*} d{\hat{k}} \, {\text {den}}[f_{T}'(h^\delta _{\mathrm{per}, 0} - \mu )] |\Omega _\delta |^{-1} {\hat{\varphi }}(k) . \end{aligned}$$

(C. 14)

where $r^\delta _{\mathrm{per, 0}}(z)$ and $h^\delta _{\mathrm{per}, 0}$ are given in (C. 6). Applying the inverse Bloch–Floquet transform (2.24), (C. 14) implies

$$\begin{aligned} M_\delta '&P_r\varphi = -\delta {\text {den}}[f_{T}'(h^\delta _{\mathrm{per}, 0}-\mu ) ] \int _{\Omega _\delta ^*} d{\hat{k}} \, e^{-ikx} |\Omega _\delta |^{-1}{\hat{\varphi }}(k). \end{aligned}$$

(C. 15)

Since $d{\hat{k}}$ is normalized by the volume $|\Omega _\delta ^*|$ (which is independent of the choice of the cell), (C. 15) shows

$$\begin{aligned} M_\delta ' P_r\varphi =&- \delta {\text {den}}\left[ f_{T}'(h^\delta _{\mathrm{per}, 0}-\mu ) \right] P_r\varphi . \end{aligned}$$

(C. 16)

By Lemma 2.7 and recalling the definition of $U_{\delta }$ from (2.45), we see that

$$\begin{aligned} \delta {\text {den}}\left[ f_{T}'(h^\delta _{\mathrm{per}, 0}-\mu ) \right]&= \delta {\text {den}}\left[ U_{\delta }f_{T}'(h_{\mathrm{per}, 0}-\mu ) U_{\delta }^* \right] \end{aligned}$$

(C. 17)

$$\begin{aligned}&= \delta ^{-2} {\text {den}}\left[ f_{T}'(h_{\mathrm{per}, 0}-\mu ) \right] (\delta ^{-1}x) \, , \end{aligned}$$

(C. 18)

where $h_{\mathrm{per}, 0} = h_{\mathrm{per}, 0}^{\delta = 1}$, which together with (C. 16) gives (C. 10)–(C. 11). $\square$

Proof of (C. 3)

Let $V_\delta$ be given in (C. 11). Since the Bloch–Floquet decomposition is unitary, we see, by Lemma C.3 and Corollary 2.4, that

$$\begin{aligned} \Vert M_\delta ' P_r\varphi \Vert _{L^2}^2 =&\Vert V_\delta P_r\varphi \Vert _{L^2}^2 = \int _{\Omega _\delta ^*} d{\hat{k}} \Vert V_\delta |{\hat{\varphi }}(k)| |\Omega _\delta |^{-1} \Vert _{L^2_{\mathrm{per}, \delta }}^2, \end{aligned}$$

(C. 19)

where $L^2_{\mathrm{per}, \delta }$ is given in (C. 1). Using the fact that $d{\hat{k}} = |\Omega _\delta ^*|^{-1} dk$ and $|\Omega _\delta | = \delta ^3 |\Omega |$, (C. 19) implies

$$\begin{aligned} \Vert M_\delta ' P_r\Vert _{L^2}^2 =&\delta ^{-3} |\Omega |\Vert V_\delta \Vert _{L^2_{\mathrm{per}, \delta }}^2 \Vert P_r\varphi \Vert _{L^2}^2. \end{aligned}$$

(C. 20)

By a change of variable, we see that $\Vert V_\delta \Vert _{L^2_{\mathrm{per}, \delta }} = \delta ^{-1/2}\Vert V \Vert _{L^2_{\mathrm{per}}},$ where V is given by (4.28). Combing with (C. 20), the fact ${\bar{P_r}} (-i\nabla )^{-1} \lesssim r^{-1}$ (where $\nabla ^{-1}$ is given in (3.56)) and $r^{-1}= a^{-1}\delta \lesssim \delta$ (see (3.38)) yields Eq. (C. 3). $\square$

Proof of (C. 4)

Let $M_\delta ''$ be given by (C. 9) and $k^{-1} := k/|k|^2$. Let $\varphi \in L^2(\mathbb {R}^3)$. By Corollary 2.4, we have

$$\begin{aligned} (P_r\nabla ^{-1} \varphi )_k= k^{-1} {\hat{\varphi }}(k) |\Omega _\delta |^{-1} \chi _{B(r)}(k). \end{aligned}$$

(C. 21)

This gives $M_\delta '' P_r\nabla ^{-1} \varphi = |\Omega _\delta |^{-1}\int ^\oplus _{B(r)} d{\hat{k}} M''_{\delta , k} k^{-1} {\hat{\varphi }}(k)$. Since the Bloch–Floquet decomposition is unitary, we see, using (C. 21), that

$$\begin{aligned} \Vert M_\delta '' P_r\nabla ^{-1}&\varphi \Vert _{L^2}^2 \nonumber \\&= |\Omega _\delta |^{-2}\int _{B_r} d{\hat{k}} \, \Vert M''_{\delta , k} 1\Vert _{L^2_{\mathrm{per}, \delta }}^2 |k|^{-2} |{\hat{\varphi }}(k)|^2. \end{aligned}$$

(C. 22)

Since $d {\hat{k}} = |\Omega ^*|^{-1} dk = |\Omega | dk$ and $|\Omega _\delta | = \delta ^3|\Omega |$, (C. 22) is bounded as

$$\begin{aligned} \Vert M_\delta '' P_r\nabla ^{-1}&\varphi \Vert _{L^2}^2 \nonumber \\&\lesssim \delta ^{-3} \sup _{k \in B_r} \left( \Vert M''_{\delta , k} 1\Vert _{L^2_{\mathrm{per}, \delta }}^2 |k|^{-2} \right) \Vert P_r\varphi \Vert _{L^2}^2, \end{aligned}$$

(C. 23)

where $1 \in L^2_{\mathrm{per}, \delta }$ is the constant function 1 and $L^2_{\mathrm{per}, \delta }$ is given in (C. 1).

By (C. 5) and (C. 7), we have, for $M''_{\delta , k}$ given in (C. 7), that

$$\begin{aligned} M''_{\delta , k} \varphi =&- \delta {\text {den}}\left[ \oint r_{\mathrm{per}, 0}(z) \varphi (r^\delta _{\mathrm{per}, k}(z)-r^\delta _{\mathrm{per}, 0}(z)) \right] . \end{aligned}$$

(C. 24)

Since $r^\delta _{\mathrm{per}, k}(z)-r_{\mathrm{per}, 0}(z)=r^\delta _{\mathrm{per}, 0}(z)A_k r^\delta _{\mathrm{per}, k}(z)$, where $A_k:=-2 (-i\nabla ) \delta k+\delta ^2 |k|^2$, this gives

$$\begin{aligned} M''_{\delta , k} \varphi =&- \delta {\text {den}}\oint \left[ r^\delta _{\mathrm{per}}(z) \varphi r^\delta _{\mathrm{per}}(z) A_k r^\delta _{\mathrm{per}, k}(z) \right] . \end{aligned}$$

(C. 25)

By the rescaling relation (3.34) and (C. 25), we see that

$$\begin{aligned} \Vert M''_{\delta , k} 1\Vert _{L^2_{\mathrm{per}, \delta }}&= \Vert U_\delta ^* M''_{\delta , k}U_\delta \cdot U_\delta ^* 1\Vert _{L^2_{\mathrm{per}}} = \delta ^{3/2-2}\Vert M''_{1, k} 1\Vert _{L^2_{\mathrm{per}}}\nonumber \\&= \delta ^{-1/2} \big \Vert {\text {den}}\big [ \oint r_{\mathrm{per}}^2(z) A_kr_{\mathrm{per}, \delta k}(z) \big ] \big \Vert _{L^2_{\mathrm{per}}}. \end{aligned}$$

(C. 26)

By (C. 26), notation $A_k:=-2 (-i\nabla ) \delta k+\delta ^2 |k|^2$ and inequality (4.32), with $\alpha =0,1/2$, we obtain, for $|k| \le r$,

$$\begin{aligned} \Vert M''_{\delta , k}1\Vert _{L^2_{\mathrm{per}}} |k|^{-1}&\lesssim \delta ^{-1/2+1} + \delta ^{-1/2+2} r \nonumber \\&= \delta ^{1/2} + \delta ^{3/2} r \, . \end{aligned}$$

(C. 27)

By (3.38) and (C. 23), Eq. (C. 27) shows that

$$\begin{aligned} \Vert M_\delta '' P_r\nabla ^{-1} \varphi \Vert _{L^2} \lesssim \delta ^{-1} . \end{aligned}$$

(C. 28)

This bound, the observation that $\Vert {\bar{P_r}} \nabla ^{-1}\Vert _\infty \lesssim r^{-1}$ (see (3.37)) and the definition $r =a/\delta > rsim 1/\delta$ imply Eq. (C. 4). $\square$

This completes the proof of Proposition C.1. $\square$

We use Proposition C.1 to prove the following

Proposition C.4

Let Assumption [A1] hold and let $\beta e^{-\eta _0\beta }\lesssim 1$ (which is weaker than Assumption [A3]). Then the operator $M_\delta$ is bounded as

$$\begin{aligned} \Vert \nabla ^{-1}{\bar{P_r}}&M_\delta f\Vert _{\dot{H}^1} \lesssim \Vert f\Vert _{\delta }. \end{aligned}$$

(C. 29)

Proof of Proposition C.4

Decomposing $M_\delta$ according to (C. 2) and using bounds Eqs. (C. 3) and (C. 4) of Proposition C.1, we see that

$$\begin{aligned} \Vert \nabla ^{-1}{\bar{P_r}} M_\delta P_r&\varphi \Vert _{L^2} \le \Vert \nabla ^{-1}{\bar{P_r}} M_\delta ' P_r\varphi \Vert _{L^2} + \Vert \nabla ^{-1}{\bar{P_r}} M_\delta '' P_r\varphi \Vert _{L^2}\nonumber \\&\lesssim \delta ^{-1} \Vert V \Vert _{L^2_{\mathrm{per}}} \Vert f\Vert _{L^2} + \Vert \nabla f\Vert _{L^2}, \end{aligned}$$

(C. 30)

where V is given in (4.28). Using $\Vert V\Vert _{L^2_{\mathrm{per}}} \le \Vert V\Vert _{L^\infty }^{1/2} \Vert V\Vert _{L_{\mathrm{per}}^1}^{1/2}$ and the definition $m:=\Vert V\Vert _{L^1_{\mathrm{per}}}$ in (C. 30) gives

$$\begin{aligned} \Vert \nabla ^{-1}{\bar{P_r}}&M_\delta f\Vert _{\dot{H}^1} \lesssim \Vert V \Vert _{L^\infty }^{1/2} (\delta ^{-1} m^{1/2}\Vert f\Vert _{L^2}) + \Vert \nabla f\Vert _{L^2} \, . \end{aligned}$$

(C. 31)

Lemma B.2 and definition (3.45) imply (C. 29). $\square$

Appendix D: Refined Nonlinear Estimates

Let $N_\delta$ be given implicitly by (3.31) and recall the definition of the $B_{s,\delta }$ norm from (3.45). Let $\dot{H}^{0}\equiv L^2$. In this section we prove estimates on $N_\delta$.

Proposition D.1

Let Assumption [A1] hold. If $\Vert \varphi _1\Vert _{B_{s,\delta }}, \Vert \varphi _2\Vert _{B_{s,\delta }} = o(\delta ^{-1/2})$, then we have the estimate

$$\begin{aligned} \Vert N_\delta&(\varphi _1) - N_\delta (\varphi _2)\Vert _{L^2} \nonumber \\&\lesssim e^{-\beta }m^{ -1/3} \delta ^{-1/2}(\Vert \varphi _1\Vert _{B_{s,\delta }} + \Vert \varphi _2\Vert _{B_{s, \delta }})\Vert \varphi _1 - \varphi _2\Vert _{\dot{H}^1} . \end{aligned}$$

(D.1)

We derive Proposition D.1 from its version with $\delta = 1$ by rescaling. For $\delta = 1$, we have the following result.

Proposition D.2

Let Assumption [A1] hold $\Vert \psi \Vert _{L^2} = o(1)$. Then $N:=N_{\delta =1}$ satisfies the estimate

$$\begin{aligned} \Vert N(\psi _1) - N(\psi _2)\Vert _{L^2}&\lesssim \sum _{j=1}^2\bigg [ (\Vert \psi _j\Vert _{\dot{H}^1})\Vert \psi _1 - \psi _2\Vert _{\dot{H}^1} \nonumber \\&\quad + e^{-\beta } \big (\Vert \psi _j\Vert _{\dot{H}^1}^{1/3} \Vert \psi _j\Vert _{L^2}^{2/3} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}} \nonumber \\&\quad + \Vert \psi _j\Vert _{\dot{H}^{1}} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}}^{1/3}\Vert \psi _1 - \psi _2\Vert _{L^2}^{2/3}\big )\bigg ]. \end{aligned}$$

(D.2)

The derivation of Proposition D.1 from Proposition D.2 is same as that of Proposition 5.1 from Proposition 5.2 and we omit it here.

Proof of Proposition D.2

Let $h_{\mathrm{per}}$ and $r_{\mathrm{per}}(z)$ be given in (3.8). We use the relations (5.6)–(5.12) in the proof of Proposition 5.2. Following the latter proof we see that it suffices to improve the estimate of $N_k(\psi )$ in Proposition 5.3, to which we proceed. $\square$

Proposition D.3

Let Assumption [A1] hold and let $N_k$ be given by (5.12). Assume that $\Vert \nabla \psi \Vert _{L^2} = o(1)$, then, for any $k\ge 2$, we have the estimate

$$\begin{aligned}&\Vert {\text {den}}[N_k(\psi )]\Vert _{L^2} \lesssim \Vert \nabla \psi \Vert _{L^2}^k + e^{-\beta } \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}\delta _{k, 2}, \end{aligned}$$

(D.3)

where the constants associated with $\lesssim$ are independent of $\beta$ and $\delta _{k, 2}$ is the Kronecker delta.

Proof

We begin with $k=2$. To improve upon estimate (5.13), we, following [8], use the partition of unity

$$\begin{aligned} P_1+ P_2= \mathbf {1}, \text { with } P_1 := \chi _{h_{\mathrm{per}} < \mu } \text { and } P_2 := \chi _{h_{\mathrm{per}} \ge \mu }. \end{aligned}$$

(D.4)

Let $R_i \equiv R_i (z)= r_{\mathrm{per}}(z) P_i$ where $i=1, 2,$$P_i$ and $r_{\mathrm{per}}(z)$ are given in (D.4) and (3.8). Recalling definition (5.12) of $N_k(\psi )$ and inserting the partition of unity, $P_1 + P_2 = \mathbf {1}$, after each R in the integrand of (5.12), we arrive at

$$\begin{aligned} N_2(\psi ) = \sum _{a,b,c = 1,2}N^{abc}_2(\psi ) \, , \end{aligned}$$

(D.5)

where the $N_2^{abc}(\psi )$, for $a,b,c =1, 2$, denote the operators

$$\begin{aligned} N^{abc}_2(\psi ) =&\oint R_a \phi R_b \psi R_c \, . \end{aligned}$$

(D.6)

We estimate the terms individually. Below, we use the estimate (see (4.32))

$$\begin{aligned} \Vert (1-\Delta )^{\alpha } R_i(z)\Vert \lesssim \&d^{\alpha -1}\lesssim 1,\ i=1, 2, \end{aligned}$$

(D.7)

for $\alpha \in [0, 1]$ and $z\in \Gamma$, where

$$\begin{aligned} d \equiv d(z):=\mathrm {dist}(z, \sigma (h_{\mathrm{per}}))\ge \frac{1}{4} . \end{aligned}$$

(D.8)

Case 1 (121) and (212). We estimate the case for (121), the other case is done similarly. Since $P_1P_2 = 0$, we write

$$\begin{aligned} N^{(121)}_2(\psi )&= \oint R_1 \psi R_2 P_2 \psi R_1 \end{aligned}$$

(D.9)

$$\begin{aligned}&= \oint R_1 [P_1, \psi ]P_2 R_2 P_2 [\psi , P_1] R_1 \, . \end{aligned}$$

(D.10)

Applying Lemma 2.1 and Eq. (D.7) to the r.h.s. and using that the operator norm is bounded by the $I^2$ norm, we find

$$\begin{aligned} \Vert {\text {den}}[N^{(121)}_2(\psi )] \Vert _{L^2}&\lesssim \Vert (1-\Delta )^{3/4+\epsilon } N^{(121)}_2(\phi )\Vert _{S^2} \end{aligned}$$

(D.11)

$$\begin{aligned}&\lesssim \left| \oint \right| d^{-3} \Vert [P_1, \psi ]P_2 \Vert _{S^2}^2. \end{aligned}$$

(D.12)

where, recall,

$$\begin{aligned} \left| \oint \right| :=&\frac{1}{2\pi } \int _\Gamma dz |f_{T} (z-\mu )|. \end{aligned}$$

(D.13)

A key observation allowing us to obtain an improved estimate is that the commutators lead to gradient estimates:

Lemma D.4

Let Assumption [A1] hold, we have the estimate

$$\begin{aligned} \Vert [P_i, \psi ] \Vert _{S^2} \lesssim \Vert \nabla \psi \Vert _{L^2},\ i=1, 2 . \end{aligned}$$

(D.14)

Proof of Lemma D.4

Since the identity commutes with any operator and $P_2 = \mathbf {1}- P_1$ (see (D.4)), we prove the lemma for $P_1$ only. Since $h_{\mathrm{per}}$ (see (3.8)) has a gap at $\mu$, the Cauchy integral formula implies

$$\begin{aligned} P_1 = \frac{1}{2\pi i} \int _{\Gamma _1} (z-h_{\mathrm{per}})^{-1} = \frac{1}{2\pi i} \int _{\Gamma _1} r_{\mathrm{per}}(z) \end{aligned}$$

(D.15)

where $\Gamma _1$ is the contour $\{t + i ; -c\le t< \mu \} \cup \{t - i ; -c\le t < \mu \} \cup \{ -c -it + (1-t)i : t \in [0,1] \} \cup \{ \mu -it + (1-t)i : t \in [0,1] \}$, where $c > 0$ is any constant such that $h_{\mathrm{per}} > -c +1$, and the contour is traversed counter-clockwise. We see that

$$\begin{aligned}{}[P_1, \psi ]&= \frac{1}{2\pi i} \int _{\Gamma _1} [ r_{\mathrm{per}}(z) ,\psi ] \end{aligned}$$

(D.16)

$$\begin{aligned}&= \frac{1}{2\pi i} \int _{\Gamma _1} r_{\mathrm{per}}(z) [\nabla \cdot , \nabla \psi ] r_{\mathrm{per}}(z) \nonumber \\&\quad + \frac{1}{2\pi i} \int _{\Gamma _1} r_{\mathrm{per}}(z) (2\nabla \psi \cdot \nabla ) r_{\mathrm{per}}(z). \, \end{aligned}$$

(D.17)

Lemma D.4 is now proved by an application of the Kato–Seiler–Simon inequality ((2.14)) to (D.17) and noting that $\Gamma _1$ is compact and has length O(1). $\square$

Using Lemma D.4 and estimates (3.6) and (D.8) in (D.12) yields that

$$\begin{aligned} \Vert {\text {den}}[N^{(121)}_2(\psi )] \Vert _{L^2} \lesssim \Vert \nabla \psi \Vert _{L^2}^2 \, . \end{aligned}$$

(D.18)

Case 2: (112), (211), (122), (221) We estimate the case for (112), the other cases are done similarly. Again, since $P_1P_2 = 0$, we write

$$\begin{aligned} N^{(112)}_2(\psi )&= \oint R_1 \psi R_1 \psi R_2 \end{aligned}$$

(D.19)

$$\begin{aligned}&= \oint R_1 \psi R_1 [\psi , P_1] R_2 \, . \end{aligned}$$

(D.20)

Using Lemma 2.1 as in with $N^{(121)}_2(\psi )$ in (D.11), we estimate (D.20) as

$$\begin{aligned} \Vert {\text {den}}[N^{(112)}_2(\psi )] \Vert _{L^2}&\lesssim \left| \oint \right| d^{-1} \Vert \psi R_1 \Vert \Vert [\psi , P_1]R_2 \Vert _{S^2} . \end{aligned}$$

(D.21)

where $\left| \oint \right|$ is defined in (D.13). By the inequality $\Vert A\Vert \le \Vert A\Vert _{I^p}$, for any $p < \infty$ for any operator A on $L^2(\mathbb {R}^3)$, and the Kato–Seiler–Simon inequality (2.14), we find $\Vert \psi R_1 \Vert \le \Vert \psi R_1 \Vert _{S^6}\lesssim \Vert \psi \Vert _{L^6}$. Using this, together with Lemma D.4, in (D.21), we obtain

$$\begin{aligned} \Vert {\text {den}}[N^{(112)}_2(\psi )] \Vert _{L^2} \lesssim&\left| \oint \right| d^{-2} \Vert \psi \Vert _{L^6} \Vert \nabla \psi \Vert _{L^2} . \end{aligned}$$

(D.22)

Combining this with (3.6), (D.8) and Hardy–Littlewood’s inequality (2.18) gives

$$\begin{aligned} \Vert {\text {den}}[N^{(112)}_2(\psi )] \Vert _{L^2} \lesssim&\Vert \nabla \psi \Vert _{L^2}^2. \end{aligned}$$

(D.23)

Case 3 (111) and (222). We use the $L^2$–$L^2$ duality to estimate the $L^2$ norm of ${\text {den}}[N^{(qqq)}_2(\psi )], q=1, 2$. We have, by (2.5) and definition (5.12),

$$\begin{aligned} \Vert {\text {den}}[N^{(qqq)}_2(\psi )]\Vert _{L^2}&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \int f {\text {den}}[N^{(qqq)}_2(\psi )]\right| \nonumber \\&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \mathrm {Tr}[f N^{(qqq)}_2(\psi )]\right| \nonumber \\&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \oint \mathrm {Tr}(f R_q \psi R_q \psi R_q)\right| . \end{aligned}$$

(D.24)

(In the last two lines, f is considered as a multiplication operator.) To show that the integral on the r.h.s. converges absolutely, we follow the arguments in (5.17)–(5.20) to prove, for $q=0, 1$,

$$\begin{aligned} \big |\mathrm {Tr}( f R_q (\psi R_q)^2)\big |\lesssim&d^{-4/3}\Vert f\Vert _{L^2} \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}. \end{aligned}$$

(D.25)

Due to definition (D.8) of $d \equiv d(z)$, this shows that the integral on the r.h.s. of (D.24) converges absolutely.

Lemma D.5

For $q=1, 2$, we have

$$\begin{aligned} \oint&\mathrm {Tr}[f R P_q g R P_q h P_q] \nonumber \\&= \frac{1}{2\pi i}\int _{\Gamma _q} (f_{T}(z) - 1) \mathrm {Tr}[f R P_q g R P_q h P_q]. \end{aligned}$$

(D.26)

Proof

Note that the contour $\Gamma$ in Fig. 2 is the union of two disjoint contours, $\Gamma =\Gamma _1 \cup \Gamma _2$, with $\Gamma _1$ being the closed contour and $\Gamma _1$ unbounded one (i.e. the parts of $\Gamma$ with $\mathrm {Re}\, z < \mu$ and $\mathrm {Re}\, z > \mu$). We first note that, by Bloch’s theory,

$$\begin{aligned} \int _{\Gamma _q} dz \,&\mathrm {Tr}[f R P_q g R P_q h P_q] =\int _{\Gamma _q} dz\, \int _{(\Omega ^*)^3} d{\hat{k}} d{\hat{k}}_1 d {\hat{k}}_2 \end{aligned}$$

(D.27)

$$\begin{aligned}&\times \mathrm {Tr}_{L^2_{\mathrm{per}}} f_{k-k_1} (RP_q)_{k_1} g_{k_1-k_2} (RP_q)_{k_2} h_{k_2-k} (RP_q)_k . \end{aligned}$$

(D.28)

Computing the trace in the complete orthonormal basis of eigenvectors $\varphi _{m,k}$ of $(RP_q)_{k}$ (with eigevalues $\lambda _{m,k}$) and inserting the complete orthonormal bases of eigenvectors $\varphi _{n,k_1}$ and $\varphi _{r, k_2}$ of $(RP_q)_{k_1}$ and $(RP_q)_{k_2}$ (with eigevalues $\lambda _{n,k_1}$ and $\lambda _{r,k_2}$) into (D.28), we see that

$$\begin{aligned} \int _{\Gamma _1}&\mathrm {Tr}[f R P_q g R P_q h P_q] = \sum _{m,n,r} \int _{(\Omega ^*)^3} d{\hat{k}} d{\hat{k}}_1 d {\hat{k}}_2 \end{aligned}$$

(D.29)

$$\begin{aligned}&\times \langle \varphi _{m,k}, f_{k-k_1} \varphi _{n, k_1} \rangle \langle \varphi _{n, k_1}, g_{k_1-k_2} \varphi _{r, k_2} \rangle \langle \varphi _{r, k_2}, h_{k_2-k} \varphi _{m, k} \rangle \end{aligned}$$

(D.30)

$$\begin{aligned}&\times \int _{\Gamma _q} dz\, \frac{1}{(z-\lambda _{m,k})(z-\lambda _{n,k_1})(z-\lambda _{r, k_2})} . \end{aligned}$$

(D.31)

Since $P_1$ projects to the spectrum of $h_{\mathrm{per}}$ on the left of $\mu$, we see that $\lambda _{m,k}, \lambda _{n,k_1}, \lambda _{r, k_2} < \mu$. In particular, these eigenvalues are in the left closed contour in Fig. 2. Consequently, Cauchy’s integral formula shows that the term in the large bracket in (D.31) is identically zero. Similar argument applies to $P_2$. This shows that

$$\begin{aligned} \frac{1}{2\pi i}\int _{\Gamma _q} \mathrm {Tr}[f R P_q g R P_q h P_q] = 0. \end{aligned}$$

(D.32)

Thus (D.26) follows. $\square$

Using the explicit form of the Fermi–Diract distribution $f_{T}$ in (1.2), we see that

$$\begin{aligned} |f_{T}(z-\mu ) - 1| = \frac{e^{\beta (\mathrm {Re}\, z-\mu )}}{|1+e^{\beta (z-\mu )}|} . \end{aligned}$$

(D.33)

By condition (1.26) and by the choice of the contour, $\Gamma$, in Fig. 2, we see that the if $z \in \Gamma$, then $\mathrm {Re}\, z$ is at least at the distance $\ge 1$ from $\mu$. Hence, for z in a contour $\Gamma$, (D.33) implies that

$$\begin{aligned} |f_{T}(z-\mu ) - 1| \lesssim e^{-\beta }. \end{aligned}$$

(D.34)

Applying estimates (D.34) and (D.25) to the r.h.s. of (D.26) and recalling the definition (D.8) of $d\equiv d(z)\ge \frac{1}{4}$,

we arrive at the inequality

$$\begin{aligned}&\left| \oint \mathrm {Tr}[f R P_q g R P_q h P_q]\right| \nonumber \\&\quad \lesssim e^{-\beta }\Vert f\Vert _{L^2} \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}. \end{aligned}$$

(D.35)

This inequality, together with the relation (D.24), gives

$$\begin{aligned} \left\| {\text {den}}[N^{(qqq)}_2(\phi )]\right\| _{L^2} \lesssim&e^{-\beta } \Vert \nabla \phi \Vert _{L^2}^{4/3} \Vert \phi \Vert _{L^2}^{2/3}. \end{aligned}$$

(D.36)

Inequalities (D.18), (D.23) and (D.36) imply estimate (D.3) for $k=2$.

Now we estimate $N_k$ for $k>2$. By (3.6) and (3.10), it suffices to estimate ${\text {den}}[R(\phi R)^k]$ where $R = r_{\mathrm{per}}(z)$ is given in (3.8). Using Lemma 2.1, we see that

$$\begin{aligned} \Vert {\text {den}}[R(\phi R)^k]\Vert _{L^2} \lesssim&\Vert (1-\Delta )^{4/3+\epsilon }R(\phi R)^k\Vert _{S^2} \end{aligned}$$

(D.37)

$$\begin{aligned} \lesssim&\Vert (\phi R)^k\Vert _{S^2}. \end{aligned}$$

(D.38)

Using Hölder’s inequality with $\frac{1}{2} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} +$ another k terms of $\frac{1}{\infty }$, (D.38) becomes

$$\begin{aligned} \Vert {\text {den}}[R(\phi R)^k]\Vert _{L^2} \le&\Vert \phi R\Vert _{S^6}^3 \Vert \phi R\Vert ^{k-3} \end{aligned}$$

(D.39)

$$\begin{aligned} \le&\Vert \phi R\Vert _{S^6}^k , \end{aligned}$$

(D.40)

where the last line follows since $\Vert \cdot \Vert \le \Vert \cdot \Vert _{S^p}$ for $p < \infty$. Combining with Kato–Seiler–Simon’s inequality (2.14) and Hardy-Littlewood’s inequality (2.18), (D.40) implies (D.3) for $k\ge 3$. $\square$

The rest of the proof of Proposition D.1 proceeds as the proof of in Proposition 5.3. $\square$