Let $\mathcal {L}$ be a finite lattice with largest element $X$ and $\mathcal {A}$ a $C^*$-algebra. We say that $\mathcal {A}$ is $\mathcal {L}$-graded if a family $\{\mathcal {A}(Y)\}_{Y\in \mathcal {L}}$ of $C^*$-subalgebras has been given such that $\mathcal {A} = \sum _{y\in \mathcal {L}}\mathcal {A}(Y)$ (direct sum) and $\mathcal {A}(Y)\mathcal {A}(Z) \subset \mathcal {A}(Y \vee Z)$ for $Y, Z \in \mathcal {L}$. The Hamiltonians usually considered in the many-body problems are affiliated to such an algebra. If $\mathcal {A}$ is realized on a Hilbert space $\mathcal {H}$, the many-channel structure of a self-adjoint operator $H$ (in general non densely defined) affiliated to $\mathcal {A}$ may be described as follows : for each $Y \in \mathcal {L}, \mathcal {A}(Y) = \sum _{Z\leq Y} \mathcal {A}(Z)$ is a $C^*$-algebra, the natural projection $\mathcal {P}_Y: \mathcal {A} \to \mathcal {A}_Y$ is a $\ast $-homomorphism and there is a unique self-adjoint operator $H_Y$ such that $\mathcal {P}_Y(f(H)) = f(H_Y)$ for all $f \in C^\infty (\mathbf {R})$. Let $A$ be a self-adjoint operator such that $e^{-iA\alpha }\mathcal {A}(Y)e^{iA\alpha } \subset \mathcal {A}(Y)$ for all $Y$ and $\alpha $. Assume that $D(H_Y)$ is invariant under $e^{iA\alpha }$ for all $Y$ and $\frac {d}{d\alpha }e^{-iA\alpha }He^{iA\alpha }$ exists in norm in $B(D(H),D(H)^*)$ and $H$ has a spectral gap. Our main result is that, under a further assumption on $\mathcal {A}$ which is independent of $H$ and trivially verified in the $N$-body case. $A$ is conjugate to $H$ at a point $\lambda \in \mathbb {R}$ if it is conjugate to each $H_Y$ with $Y \neq X$ at $\lambda $.