Assume $H$ is a Hilbert space and $a_1,\dots,a_n$ are operators with Hermitian adjoints $a_1^*,\dots,a_n^*$, satisfying the canonical commutation relations. Define $N_j=a_j^*a_j$. Assume $v$ is an eigenvector of $\sum N_j$.
Is $v$ an eigenvector of each $N_j$? How can we see that a common eigenvector exists for the $N_j$'s?
Edit: The canonical commutation relations are $[a_i,a_j]=[a^*_i,a^*_j]=0, [a_i,a^*_j]=\delta_j^i$