Let $Q: \mathbb{R} \rightarrow \mathbb{R}$ be a $C_c^{\infty}$ function, i.e. it is smooth and has c ompact support. We assume $Q$ is even, i.e. $Q(x)=Q(-x)$. We assume $Q$ is non-trivial,(i.e. $Q$ does not equal to zero ever ywhere).
Let $T_1(x):=x Q(x)$, and let $T_2(x)=x^2 Q(x)$. Let $T_3:=e^{-x^2}\left(1+x^{2024}\right)$
We also introduce the following notation. For any $f: \mathbb{R} \rightarrow \mathbb{R}, \lambda>0, \alpha \in \mathbb{R}$, we define
$$
f_{\lambda, \alpha}:=\frac{1}{\lambda^{1 / 2}} f\left(\frac{x-\alpha}{\lambda}\right)
$$
We claim: There exists $\delta>0, \varepsilon>0$, so that for any $c \in \mathbb{R}$ wit $\mathrm{h}|c| < \delta$, one can find unique $\lambda, \alpha$ such that the following $\mathrm{h}$ old
(1) $|\lambda-1|+|\alpha| < \varepsilon$.
(2) $\left\langle Q_{\lambda, \alpha}-Q-c T_3, T_1\right\rangle=0$.
(3) $\left\langle Q_{\lambda, \alpha}-Q-c T_3, T_2\right\rangle=0$.
(Here, for any two functions $f_1, f_2$, we define $\left.\left\langle f_1, f_2\right\rangle:=\int f_1(x) f_2(x) \mathrm{d} x\right)$.
Is the above claim correct? Prove your conclusion.