I have a question with a theorem which appears in a text called “Invitation to Random Schrödinger Operators”, in unit 7.
Theorem 7.7. Let $H$ be a selfadjoint operator on Hilbert Space, take $\psi \in H_{pp}$ and let $\Lambda_L$ denote a cube in $\mathbb Z^d$ centered at the origin with side length $2L+1$. Then:
(1) $$\lim_{L \to \infty} \sup_{t \geq 0} \left(\sum_{x \in \Lambda_L} |e^{-itH} \psi(x)|^2 \right) = ||\psi||^2$$
and
(2) $$\lim_{L \to \infty} \sup_{t \geq 0} \left(\sum_{x \notin \Lambda_L} |e^{-itH} \psi(x)|^2 \right) = 0 $$
Proof: Since $e^{-itH}$ is unitary, we have for all $t$
$$||\psi||^2 =||e^{-itH} \psi||^2 = \sum_{x \in \Lambda_L } |e^{-itH} \psi(x)|^2 + \sum_{x \notin \Lambda_L} |e^{-itH}\psi(x)|^2 $$
Above we saw that the expression is valid for eigenfunctions $\psi$. To prove it for other vectors in $H_{pp}$ we introduce the following notation: By $P_L$ we denote the projection onto $C\Lambda_L$. Then equation (2) claims that:
$$ ||P_L e^{-itH} \psi||^2 \to 0$$
uniformly in $t$ as $L \to \infty$. If $\psi$ is a finite linear combination of eigenfunctions, say $\psi=\sum_{m=1}^M \alpha_k \psi_k$, $H\psi_k=E_k \psi_k$, then:
(3) $$ ||P_L e^{-itH} \psi || = || \sum_{m=1}^M \alpha_k P_L e^{-itH} \psi_k || \leq \sum_{m=1}^M |\alpha_k| ||P_L e^{-itH} \psi_k || = \sum_{m=1}^M |\alpha_k| ||P_L e^{-itE_k} \psi_k|| = \sum_{m=1}^M |\alpha_k| ||P_L \psi_k||$$
From this moment, it is that I begin to have problems to understand
By taking $L$ large enough, each term in the sum above can be made smaller then $(\sum_{m=1}^M |\alpha_k|)^{-1} \varepsilon$. If now $\psi$ is an arbitrary element of $H_{pp}$ , there is a linear combination of eigenfunctions $\psi^{(M)} = \sum_{m=1}^M \alpha_k \psi_k$ such that $||\psi-\psi^{(M)}|| < \varepsilon $ $||P_L e^{-itH} \psi|| \leq ||P_L e^{-itH} \psi^{(M)} || + ||P_L e^{-itH} (\psi - \psi^{(M)}) || \leq ||P_L e^{-itH} \psi^{(M)} || + || \psi - \psi^{(M)}||$
GUYS I can’t understand how $||P_L e^{-itH} (\psi - \psi^{(M)} ) || \leq || \psi - \psi^{(M)}||$
Thanks!