When I read YVES COLIN DE VERDIÈRE's paper: Sur la multiplicité de la première valeur propre non nulle du laplacien, he gave a proposition without proof:
$\mathcal{H}$ is a Hilbert space, $Q$ is a positive closed quadratic form on $\mathcal{H}$ with domain $D(Q)$. If $D(Q)$ admits the $Q$-orthogonal decomposition $D(Q)=\mathcal{H}_0\oplus\mathcal{H}_\infty$ and $\forall x\in \mathcal{H}_\infty$, $Q(x)\geq C^2|x|^2$. Then there exists a universal constant $C_N>0$ such that if $x\in \oplus(\mathcal{H}_{i_1}\otimes\cdot\cdot\cdot\otimes \mathcal{H}_{i_N})$ where $i_l\in\{0,\infty\}$ and there exists $l$ such that $i_l=\infty$, we have $Q^{\otimes^N}(x)\geq C_NC^2|x|^2_{\otimes^N}$.
As a quadratic form, $Q^{\otimes^N}(\varphi_1\otimes\cdot\cdot\cdot\otimes\varphi_N)=Q(\varphi_1)|\varphi_2|^2\cdot\cdot\cdot|\varphi_N|^2+|\varphi_1|^2Q(\varphi_2)\cdot\cdot\cdot|\varphi_N|^2+\cdot\cdot\cdot+|\varphi_1|^2\cdot\cdot\cdot|\varphi_{N-1}|^2Q(\varphi_N)$.
I think this proposition may be wrong and I construct a counterexample： $\mathcal{H}=\mathbb{R}^2$, The matrix representation of $Q$ is \begin{pmatrix}1\quad b\\b \quad b^2\end{pmatrix} $\mathcal{H}_\infty=\mathbb{R}\cdot(1,0), \mathcal{H}_0=\mathbb{R}\cdot (b,-1)$, then $C^2=1$. For simplify, we denote $\varphi_\infty=(1,0),\varphi_0=(b,-1)$ and consider the case $N=2$. For fixed positive integer $k\geq3$, let $b=\sqrt{\frac{(2k-1)(k-1)}{k}}, \alpha=-\sqrt{\frac{(2k-4)b^2+2k-2}{2k-1}},x=\varphi_0\otimes\varphi_\infty+\varphi_\infty\otimes\varphi_0+\alpha\varphi_\infty\otimes\varphi_\infty$. By calculation, $Q^{\otimes^2}(x)=|x|_{\otimes^2}^2/k$. Since $k$ can be very large, this contradicts to the proposition.
I really confused, because it's a very famous paper, could you tell me is he wrong or I misunderstood him. I will be very grateful!!!
