Thank You for the comment. Does this result hold also for a bounded domain? What happens when $V$ changes its sign?

Which explicit formula do You have in mind? From your reasoning $||R(z,A_p)||_{\mathcal{L}(L_p)}\leq ||||G(x,t;z)||_{L_p(dx)}||_{L_{p'}(dt)}$. Now if $p=2$ one has $||||G(x,t;z)||_{L_p(dx)}||_{L_{p'}(dt)}=||G(x,t;z)||_{L_2(dx\otimes dt)}=(\sum_{n=0}^{\infty}1/|\lambda_n-z|^2)^{1/2}$, so the estimate seems to be a bit to coarse even for $p=2$.

For me it is not clear. If $z>0$ you need to show that $\sum_{n=0}^{\infty}\frac{z}{z+n^2}$ is bounded in $z$ which is clearly not.