Lie–Schwinger Block-Diagonalization and Gapped Quantum Chains with Unbounded Interactions

Abstract

We study quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian consisting of a sum of on-site terms that do not couple the degrees of freedom located at different sites of the chain and have a strictly positive energy gap above their ground-state energy. For interactions that are form-bounded w.r.t. the on-site terms, we prove that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant. Our proof is based on an extension of a novel method introduced in [FP] involving local Lie–Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.

A Appendix

Lemma A.1

For any $1\le n \le N$

$$\begin{aligned} \sum _{i=1}^{n} P^{\perp }_{\Omega _i} \ge \mathbb {1}-\bigotimes _{i=1}^{n} P_{\Omega _i}=:\,\Big (\bigotimes _{i=1}^{n} P_{\Omega _i}\Big )^{\perp } \end{aligned}$$

(4.78)

where $P^{\perp }_{\Omega _i}=\mathbb {1}-P_{\Omega _i}$.

Proof

Though the Hibert spaces ${\mathcal {H}}_j$ in (1.1) are possibly infinite dimensional, the proof of this lemma can be carried out as in Lemma A.1 of [FP]. $\quad \square$

From Lemma A.1 we derive the following bound.

Corollary A.2

For $i+r\le N$, we define

$$\begin{aligned} P^{(+)}_{I_{r,i}}:=\Big (\bigotimes _{k=i}^{i+r}P_{\Omega _{k}}\Big )^{\perp }\,. \end{aligned}$$

(A.1)

Then, for $1\le l \le L \le N-r$,

$$\begin{aligned} \sum _{i=l}^{L}P^{(+)}_{I_{r,i}}\le (r+1) \sum _{i=l}^{L+r} P^{\perp }_{\Omega _i} \,. \end{aligned}$$

(A.2)

Proof

From Lemma A.1 we derive

$$\begin{aligned} \sum _{j=i}^{i+r} P^{\perp }_{\Omega _{j}} \ge \Big (\bigotimes _{k=i}^{i+r}P_{\Omega _{k}}\Big )^{\perp }\,. \end{aligned}$$

(A.3)

By summing the l-h-s of (A.4) for i from l up to L, for each j we get not more than $r+1$ terms of the type $P^{\perp }_{\Omega _{j}}$ and the inequality in (A.3) follows. $\quad \square$

Lemma A.3

For $t>0$ sufficiently small as stated in Corollary 2.8, the following bound holds

$$\begin{aligned} (\Phi , P^{(+)}_{I_{k,q}}(G_{I_{k,q}}-E_{I_{k,q}})P^{(+)}_{I_{k,q}}\Phi ) \ge \frac{\Delta _{I_{k,q}}}{2} (\Phi , P^{(+)}_{I_{k,q}}(H^0_{I_{k,q}}+1) P^{(+)}_{I_{k,q}}\Phi ) \end{aligned}$$

(A.4)

for any vector $\Phi$ in the domain of $H^0_{I_{k,q}}$, where $\Delta _{I_{k,q}}$ is the bound⁷ from below of the spectral gap determined in Corollary 2.8. Consequently,

$$\begin{aligned} \left\| \frac{1}{(G_{I_{k,q}}-E_{I_{k,q}})^{\frac{1}{2}}}P^{(+)}_{I_{k,q}} (H^0_{I_{k,q}}+1)^{\frac{1}{2}}\right\| \le \frac{\sqrt{2}}{\Delta _{I_{k,q}}^{\frac{1}{2}}} \end{aligned}$$

(A.5)

and

$$\begin{aligned} \left\| \frac{1}{(G_{I_{k,q}}-E_{I_{k,q}})}P^{(+)}_{I_{k,q}} (H^0_{I_{k,q}}+1)^{\frac{1}{2}}\right\| \le \frac{\sqrt{2}}{\Delta _{I_{k,q}}}\,. \end{aligned}$$

(A.6)

Proof

The proof of (A.5) follows from inequality (2.45) (stated in Lemma 2.6) and (1.4). Regarding the operator norm in (A.6), we estimate

$$\begin{aligned} \left\| (H^0_{I_{k,q}}+1)^{\frac{1}{2}}P^{(+)}_{I_{k,q}} \frac{1}{(G_{I_{k,q}}-E_{I_{k,q}})^{\frac{1}{2}}}\Psi \right\| ^2 \end{aligned}$$

(A.7)

for vectors $\Psi$ of the form $\frac{(G_{I_{k,q}} -E_{I_{k,q}})^{\frac{1}{2}}\Phi }{\Vert (G_{I_{k,q}}-E_{I_{k,q}})^{\frac{1}{2}}\Phi \Vert }$, where $\Phi$ is in the domain of $G_{I_{k,q}}P^{(+)}_{I_{k,q}} =P^{(+)}_{I_{k,q}}G_{I_{k,q}}P^{(+)}_{I_{k,q}}$. The squared norm in (A.8) is seen to coincide with the l-h-s of

$$\begin{aligned} \frac{(\Phi , (H^0_{I_{k,q}}+1)\Phi )}{(\Phi , (G_{I_{k,q}} -E_{I_{k,q}})\Phi )}\le \frac{2}{\Delta _{I_{k,q}}}\,, \end{aligned}$$

(A.8)

where the inequality above corresponds to (A.5). The operator norm in (A.7) follows from (A.6) and Corollary 2.6 which implies

$$\begin{aligned} \Vert \frac{1}{P^{(+)}_{I_{k,q}}(G_{I_{k,q}}-E_{I_{k,q}})^{\frac{1}{2}} P^{(+)}_{I_{k,q}}}P^{(+)}_{I_{k,q}}\Vert \le \frac{1}{\Delta _{I_{k,q}}^{\frac{1}{2}}}. \end{aligned}$$

$\square$

Lemma A.4

Assume that $t>0$ is sufficiently small, $\Vert V^{(k,q-1)}_{I_{r,i}}\Vert _{H_0} \le t^{\frac{r-1}{4}}$, and $\Delta _{I_{k,q}}\ge \frac{1}{2}$. Then, for arbitrary N, $k\ge 1$, and $q\ge 2$, the inequalities

$$\begin{aligned}&\Vert V^{(k,q)}_{I_{k,q}}\Vert _{H_0}\le 2\Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H_0}\, \end{aligned}$$

(A.9)

$$\begin{aligned}&\Vert S_{I_{k,q}}\Vert \le At\, \Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H_0} \end{aligned}$$

(A.10)

$$\begin{aligned}&\Vert S_{I_{k,q}}(H^0_{I_{k,q}}+1)^{\frac{1}{2}}\Vert =\Vert (H^0_{I_{k,q}}+1)^{\frac{1}{2}}S_{I_{k,q}}\Vert \le Bt\, \Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H_0} \end{aligned}$$

(A.11)

hold true for universal constants A and B. For $q=1$, $V^{(k,q-1)}_{I_{k,q}}$ is replaced by $V^{(k-1,N-k+1)}_{I_{k,q}}$ in the right side of (A.10), (A.11), and (A.12).

Proof

In the following we assume $q\ge 2$; if $q=1$ an analogous proof holds. We recall that

$$\begin{aligned} V^{(k,q)}_{I_{k,q}}:= \sum _{j=1}^{\infty }t^{j-1}(V^{(k,q-1)}_{I_{k,q}})^{diag}_j \, \end{aligned}$$

(A.12)

and

$$\begin{aligned} S_{I_{k,q}}:=\sum _{j=1}^{\infty }t^j(S_{I_{k,q}})_j\ \end{aligned}$$

(A.13)

with

$$\begin{aligned} (V^{(k,q-1)}_{I_{k,q}})_1\equiv V^{(k,q-1)}_{I_{k,q}}\,, \end{aligned}$$

and, for $j\ge 2$,

$$\begin{aligned}&(V^{(k,q-1)}_{I_{k,q}})_j\, \end{aligned}$$

(A.14)

$$\begin{aligned}&\quad :=\sum _{p\ge 2, r_1\ge 1 \dots , r_p\ge 1\, ; \, r_1+\dots +r_p=j}\frac{1}{p!}\text {ad}\,(S_{I_{k, q}})_{r_1} \Big (\text {ad}\,(S_{I_{k,q}})_{r_2}\dots (\text {ad}\,(S_{I_{k,q}})_{r_p} (G_{I_{k,q}}) )\Big ) \end{aligned}$$

(A.15)

$$\begin{aligned}&\qquad +\sum _{p\ge 1, r_1\ge 1 \dots , r_p\ge 1\, ; \, r_1+\dots +r_p=j-1} \frac{1}{p!}\text {ad}\,(S_{I_{k,q}})_{r_1}\Big (\text {ad}\,(S_{I_{k,q}})_{r_2} \dots (\text {ad}\,(S_{I_{k,q}})_{r_p}(V^{(k,q-1)}_{I_{k,q}})) \Big ),\nonumber \\ \end{aligned}$$

(A.16)

and

$$\begin{aligned} (S_{I_{k,q}})_j:=\frac{1}{G_{I_{k,q}}-E_{I_{k,q}}}P^{(+)}_{I_{k,q}}\,(V^{(k,q-1)}_{I_{k,q}})_j \,P^{(-)}_{I_{k,q}}-h.c. \end{aligned}$$

(A.17)

where $j\ge 1$.

Using (A.18) we compute

$$\begin{aligned} \text {ad}\,(S_{I_{k,q}})_{r_p}(G_{I_{k,q}})= & {} \text {ad}\,(S_{I_{k,q}})_{r_p}(G_{I_{k,q}}-E_{I_{k,q}})\nonumber \\= & {} \,[\frac{1}{G_{I_{k,q}}-E_{I_{k,q}}}P^{(+)}_{I_{k,q}} \,(V^{(k, q-1)}_{I_{k,q}})_{r_p}\,P^{(-)}_{I_{k,q}}\,,\,G_{I_{k,q}}-E_{I_{k,q}}]+h.c. \end{aligned}$$

(A.18)

$$\begin{aligned}= & {} -P^{(+)}_{I_{k,q}}\,(V^{(k,q-1)}_{I_{k,q}})_{r_p}\,P^{(-)}_{I_{k,q}} -P^{(-)}_{I_{k,q}}\,(V^{(k, q-1)}_{I_{k,q}})_{r_p}\,P^{(+)}_{I_{k,q}}\,. \end{aligned}$$

(A.19)

To begin with, we show the following inequality

$$\begin{aligned} \Vert (S_{I_{k,q}})_j\Vert \le \frac{2\sqrt{2}}{\Delta _{I_{k,q}}} \Vert (V_{I_{k,q}}^{(k, q-1)})_j\Vert _{H^0}\,, \end{aligned}$$

(A.20)

where $\Vert (V_{I_{k,q}}^{(k, q-1)})_j\Vert _{H_0}$ will turn out to be bounded in the next step. As for estimate (A.21), it follows from the following computation:

$$\begin{aligned}&\Vert (S_{I_{k,q}})_j\Vert \end{aligned}$$

(A.21)

$$\begin{aligned}&\quad \le 2\left\| \frac{1}{G_{I_{k,q}}-E_{I_{k,q}}} P^{(+)}_{I_{k,q}}\,(V^{(k,q-1)}_{I_{k,q}})_j\,P^{(-)}_{I_{k,q}}\right\| \end{aligned}$$

(A.22)

$$\begin{aligned}&\quad = 2\left\| \frac{1}{G_{I_{k,q}}-E_{I_{k,q}}}P^{(+)}_{I_{k,q}} (H_{I_{k,q}}^0+1)^{\frac{1}{2}}(H_{I_{k,q}}^0+1)^{-\frac{1}{2}} (V^{(k,q-1)}_{I_{k,q}})_j(H_{I_{k,q}}^0+1)^{-\frac{1}{2}}P^{(-)}_{I_{k,q}}\right\| \end{aligned}$$

(A.23)

$$\begin{aligned}&\quad \le 2\left\| \frac{1}{G_{I_{k,q}}-E_{I_{k,q}}}P^{(+)}_{I_{k,q}} (H_{I_{k,q}}^0+1)^{\frac{1}{2}}\right\| \Vert (V_{I_{k,q}}^{(k,q-1)})_j\Vert _{H^0} \end{aligned}$$

(A.24)

$$\begin{aligned}&\quad \le \frac{2\sqrt{2}}{\Delta _{I_{k,q}}}\Vert (V_{I_{k,q}}^{(k,q-1)})_j\Vert _{H^0}\,, \end{aligned}$$

(A.25)

where we have used (A.7) in the last inequality.

Analogously, making use of (A.6) and $(H_{I_{k,q}}^0+1)^{\frac{1}{2}}P^{(-)}_{I_{k,q}}=P^{(-)}_{I_{k,q}}$, we estimate

$$\begin{aligned} \Vert (S_{I_{k,q}})_{j}(H^0_{I_{k,q}}+1)^{\frac{1}{2}}\Vert =\Vert (H^0_{I_{k,q}}+1)^{\frac{1}{2}}(S_{I_{k,q}})_{j}\Vert \le \frac{2+\sqrt{2}}{\Delta _{I_{k,q}}} \Vert (V_{I_{k,q}}^{(k, q-1)})_{j}\Vert _{H^0}\,. \end{aligned}$$

(A.26)

Next, we want to prove that

$$\begin{aligned}&\Vert (V^{(k,q-1)}_{I_{k,q}})_j\Vert _{H^0}\nonumber \\&\quad \le \sum _{p=2}^{j}\,\frac{(2c)^p}{p!} \sum _{ r_1\ge 1 \dots , r_p\ge 1\, ; \, r_1+\dots +r_p=j}\nonumber \\&\qquad \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_1}\Vert _{H^0}\Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_2} \Vert _{H^0}\dots \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_p}\Vert _{H^0}\nonumber \\&\qquad +2\Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0} \sum _{p=1}^{j-1}\,\frac{(2c)^p}{p!} \,\sum _{ r_1\ge 1 \dots , r_p\ge 1\, ; \, r_1+\dots +r_p=j-1}\nonumber \\&\qquad \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_1}\Vert _{H^0}\Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_2} \Vert _{H^0}\dots \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_p}\Vert _{H^0}\,, \end{aligned}$$

(A.27)

where $c:= \frac{2+\sqrt{2}}{\Delta _{I_{k,q}}}(> \frac{2\sqrt{2}}{\Delta _{I_{k,q}}})$. In order to show this, we note that formula (A.15) yielding $(V_{I_{k,q}}^{(k, q-1)})_j$ contains two sums. We first deal with the second, that is

$$\begin{aligned}&\sum _{p\ge 1, r_1\ge 1 \dots , r_p\ge 1\, ; \, r_1+\dots +r_p=j-1} \frac{1}{p!}\text {ad}(S_{I_{k,q}})_{r_1}\Big (\text {ad} (S_{I_{k,q}})_{r_2}\dots (\text {ad}\,(S_{I_{k,q}})_{r_p} (V^{(k,q-1)}_{I_{k,q}}))\Big )\,. \end{aligned}$$

Each summand of the above sum is in turn a sum of $2^p$ terms which, up to a sign, are obtained by permuting the factors in the following operator product

$$\begin{aligned} (S_{I_{k,q}})_{r_1}(S_{I_{k,q}})_{r_2}\ldots (S_{I_{k,q}})_{r_p} V_{I_{k,q}}^{(k, q-1)}\,, \end{aligned}$$

with the potential $V_{I_{k,q}}^{(k, q-1)}$ allowed to appear at any position. It suffices to study only one of these terms, for the others can be treated in the same way. For instance, we can treat

$$\begin{aligned} (S_{I_{k,q}})_{r_1} V_{I_{k,q}}^{(k, q-1)} (S_{I_{k,q}})_{r_2}\ldots (S_{I_{k,q}})_{r_p}. \end{aligned}$$

Notice that

$$\begin{aligned}&\Vert (S_{I_{k,q}})_{r_1} V_{I_{k,q}}^{(k, q-1)} (S_{I_{k,q}})_{r_2} \ldots (S_{I_{k,q}})_{r_p}\Vert _{H^0} \\&\quad =\Vert (H_{I_{k,q}}^0+1)^{-\frac{1}{2}} (S_{I_{k,q}})_{r_1} (H_{I_{k,q}}^0+1)^{\frac{1}{2}} (H_{I_{k,q}}^0+1)^{-\frac{1}{2}}\\&\qquad V_{I_{k,q}}^{(k, q-1)}(H_{I_{k,q}}^0+1)^{-\frac{1}{2}} (H_{I_{k,q}}^0+1)^{\frac{1}{2}} (S_{I_{k,q}})_{r_2} \ldots (S_{I_{k,q}})_{r_p} (H_{I_{k,q}}^0+1)^{-\frac{1}{2}}\Vert \\&\quad \le \Vert V_{I_{k,q}}^{(k, q-1)} \Vert _{H^0} \Vert (S_{I_{k,q}})_{r_1} (H_{I_{k,q}}^0+1)^{\frac{1}{2}} \Vert \, \Vert (H_{I_{k,q}}^0+1)^{\frac{1}{2}} (S_{I_{k,q}})_{r_2}\Vert \ldots \Vert (S_{I_{k,q}})_{r_p}\Vert \\&\quad \le c^p \Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0} \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_1}\Vert _{H^0}\Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_2} \Vert _{H^0}\dots \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_p}\Vert _{H^0}\,, \end{aligned}$$

where (A.21) and (A.27) have been used. Putting these terms together we get the second sum of (A.28).

As for the first sum in (A.15), i.e.,

$$\begin{aligned} \sum _{p\ge 2, r_1\ge 1 \dots , r_p\ge 1\, ; \, r_1+\dots +r_p=j} \frac{1}{p!}\text {ad}\,(S_{I_{k,q}})_{r_1}\Big (\text {ad}\,(S_{I_{k,q}})_{r_2} \dots (\text {ad}\,(S_{I_{k,q}})_{r_p}(G_{I_{k,q}}))\Big )\,, \end{aligned}$$

we recall the computation in (A.19) and note that each of its summands is in turn the sum up to a sign of all permutations of

$$\begin{aligned} (S_{I_{k,q}})_{r_1}(S_{I_{k,q}})_{r_2}\ldots (S_{I_{k,q}})_{r_{p-1}}[-P_{I_{k,q}}^{(+)}(V_{I_{k,q}}^{(k,q-1)})_{r_p} P_{I_{k,q}}^{(-)}- P_{I_{k,q}}^{(-)} (V_{I_{k,q}}^{(k,q-1)})_{r_p}P_{I_{k,q}}^{(+)}]\,. \end{aligned}$$

A very minor variation of the computations above shows that the $\Vert \cdot \Vert _{H^0}$-norm of the first sum in (A.15) is bounded from above by

$$\begin{aligned} \sum _{p=2}^{j}\,\frac{(2c)^p}{p!}\sum _{ r_1\ge 1 \dots , r_p\ge 1 \, ; \, r_1+\dots +r_p=j}\,\Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_1}\Vert _{H^0} \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_2}\Vert _{H^0}\dots \Vert \,(V^{(k,q-1)}_{I_{k,q}})_{r_p} \Vert _{H^0}\,; \end{aligned}$$

here we have implicitly assumed that $c>1$, without loss of generality.

From now on, we refer to the proof of Theorem 3.2 in [DFFR]. Following the procedure in [DFFR], we assume $\Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H_0}\ne 0$ and recursively define numbers $B_j$, $j\ge 1$, by the equations

$$\begin{aligned} B_1&:=\Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0}=\Vert (V^{(k,q-1)}_{I_{k,q}})_1\Vert _{H^0}, \end{aligned}$$

(A.28)

$$\begin{aligned} B_j&:=\frac{1}{a}\sum _{k=1}^{j-1}B_{j-k}B_k\,,\quad j\ge 2\,, \end{aligned}$$

(A.29)

with $a>0$ satisfying the relation

$$\begin{aligned} e^{2c a}-1+ \left( \frac{e^{2c a}-2c a -1}{a}\right) -1=0\,. \end{aligned}$$

(A.30)

Using (A.29), (A.30), (A.28), and an induction, we derive the inequality (see Theorem 3.2 in [DFFR]) for $j\ge 2$

$$\begin{aligned} \Vert (V^{(k,q-1)}_{I_{k,q}})_j\Vert _{H^0}\le B_j \,\Big (\frac{e^{2c a}-2c a-1}{a}\Big )+2\Vert V^{(k,q-1)}_{I_{k,q}} \Vert _{H^0}\,B_{j-1}\Big (\frac{e^{2c a}-1}{a}\Big )\,. \end{aligned}$$

(A.31)

From (A.29) and (A.30) it also follows that

$$\begin{aligned} B_j\ge \frac{2B_{j-1}\Vert \,V^{(k,q-1)}_{I_{k,q}}\,\Vert _{H^0}}{a} \,\quad \Rightarrow \quad B_{j-1}\le a\frac{B_j}{2\Vert \,V^{(k,q-1)}_{I_{k,q}} \,\Vert _{H^0}}\,, \end{aligned}$$

(A.32)

which, when combined with (A.32) and (A.31), yields

$$\begin{aligned} B_j\ge \Vert \,(V^{(k,q-1)}_{I_{k,q}})_j\Vert _{H^0}\,. \end{aligned}$$

(A.33)

The numbers $B_j$ are seen to be the Taylor coefficients of the function

$$\begin{aligned} f(x):=\frac{a}{2}\cdot \left( \,1-\sqrt{1- (\frac{4}{a} \cdot \Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0}) \,x }\,\right) \,, \end{aligned}$$

(A.34)

(see [DFFR]). We observe that

$$\begin{aligned} \Vert (V^{(k,q-1)}_{I_{k,q}})^{diag}_j \Vert _{H^0}= & {} \max \{\Vert P^{(+)}_{I_{k, q}}(V^{(k,q-1)}_{I_{k,q}})_j P^{(+)}_{I_{k,q}}\Vert _{H^0}\,,\,\Vert P^{(-)}_{I_{k,q}}(V^{(k,q-1)}_{I_{k,q}})_j P^{(-)}_{I_{k,q}}\Vert _{H^0}\} \end{aligned}$$

(A.35)

$$\begin{aligned}= & {} \max _{\#=\pm }\,\Vert (\frac{1}{H^0_{I_{k,q}}+1})^{\frac{1}{2}}P^{(\#)}_{I_{k,q}} (V^{(k,q-1)}_{I_{k,q}})_j P^{(\#)}_{I_{k,q}} (\frac{1}{H^0_{I_{k,q}}+1})^{\frac{1}{2}}\Vert \end{aligned}$$

(A.36)

$$\begin{aligned}= & {} \max _{\#=\pm }\,\Vert P^{(\#)}_{I_{k,q}}(\frac{1}{H^0_{I_{k,q}}+1})^{\frac{1}{2}} (V^{(k,q-1)}_{I_{k,q}})_j (\frac{1}{H^0_{I_{k,q}}+1})^{\frac{1}{2}} P^{(\#)}_{I_{k,q}}\Vert \end{aligned}$$

(A.37)

$$\begin{aligned}\le & {} \Vert (V^{(k,q-1)}_{I_{k,q}})_j \Vert _{H^0}\,. \end{aligned}$$

(A.38)

Therefore (if we consider the norms $\Vert (V^{(k,q-1)}_{I_{k,q}})^{diag}_j \Vert _{H^0}$ as t-independent) the radius of analyticity, $t_0$, of the series

$$\begin{aligned} \sum _{j=1}^{\infty }t^{j-1}\Vert (V^{(k,q-1)}_{I_{k,q}})^{diag}_j \Vert _{H^0}=\frac{1}{t}\,\Big (\sum _{j=1}^{\infty }\,t^{j}\,\Vert (V^{(k,q-1)}_{I_{k,q}})^{diag}_j \Vert _{H^0}\Big ) \end{aligned}$$

(A.39)

is bounded below by the radius of analyticity of $\sum _{j=1}^{\infty }x^jB_j$, i.e.,

$$\begin{aligned} t_0\ge \frac{a}{4\Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0}}\ge \frac{a}{4} \end{aligned}$$

(A.40)

where we have assumed $0<t<1$ and invoked the assumption $\Vert V^{(k,q-1)}_{I_{r,i}}\Vert _{H^0}\le t^{\frac{r-1}{4}}$. The same bound holds true for the radius of convergence of the series $S_{I_{k,q}}:=\sum _{j=1}^{\infty }t^j(S_{I_{k,q}})_j\,$, due to inequality (A.21) . By using (A.29) and (A.34), for $0<t<1$ and in the interval $(0,\frac{a}{8})$, we get the bound

$$\begin{aligned} \sum _{j=1}^{\infty }t^{j-1}\Vert (V^{(k,q-1)}_{I_{k,q}})^{diag}_j \Vert _{H^0}\le & {} \frac{1}{t}\sum _{j=1}^{\infty }t^jB_j \end{aligned}$$

(A.41)

$$\begin{aligned}= & {} \frac{1}{t}\cdot \frac{a}{2}\cdot \left( \,1-\sqrt{1- (\frac{4}{a}\cdot \Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0}) \,t }\,\right) \end{aligned}$$

(A.42)

$$\begin{aligned}\le & {} (1+C_a \cdot t )\,\Vert V^{(k,q-1)}_{I_{k,q}}\Vert _{H^0} \end{aligned}$$

(A.43)

for some a-dependent constant $C_a>0$. Thus we conclude that the inequality in (A.10) holds true, provided $t>0$ is sufficiently small, independently of N, k, and q. In a similar way, we derive (A.11) and (A.12), using (A.22)–(A.26) and (A.27), respectively. $\quad \square$