Direct integral representation of continuous part of Schrödinger operators / hydrogen atom

Question

Consider the operator of an idealized hydrogen atom on the Hilbert space $L^2(\mathbb R^3)$, i.e. $H = -\Delta - \frac{1}{\| \cdot \|}$ on the domain of the Laplace operator. It is well-known that it is self-adjoint and the essential spectrum is $[0,\infty)$. Let $P_+$ be the spectral projection of this operator to $[0,\infty)$. By the spectral theorem I can write the operator as multiplication by the identity on a direct integral. My question is: Can one write $P_+ H P_+$, i.e., the continuous part of $H$, as direct integral with constant fibers? I mean as an operator $T$ on $\int_{[0,\infty)}^{\oplus} \mathfrak h \ d \mu(\lambda)$ for some Hilbert space $\mathfrak h$ and some measure $\mu$ on $[0,\infty)$, given by $(T s)(\lambda) = \lambda s(\lambda)$?