Understanding the spectral measure of unbounded self-adjoint operator

Question

Let $\mathcal{H}$ be a Hilbert space and $$\Delta: D(\Delta) \rightarrow \mathcal{H}$$ a (unbounded) self-adjoint operator, densely defined on $D(\Delta) \subset \mathcal{H}$. Denote by $\mu^{\Delta}$ the spectral measure of $\Delta$. For a Borel set $E \subset \mathbb{R}$, the spectral subspace $V_E$ of $\mathcal{H}$ may be defined as $$ V_E = Range(\mu^{\Delta}(E)). $$ Let $(\lambda_n)_n$ be a real-valued sequence such that $\lambda_n \rightarrow \infty$ for $n \rightarrow \infty$. Define $$ V_n := V_{(-\lambda_n, + \lambda_n)} $$ and let $P_n$ be the orthogonal projection onto $V_n$ for $n \in \mathbb{N}$. The question is: Does $\Delta P_n$ converge to $\Delta$ in strong operator norm, i.e. for $f \in D(\Delta)$ $$ \label{1} \|\Delta P_n f - \Delta f\| \rightarrow 0. $$ My thought was following: We could write $P_n$ as $\chi_{(-\lambda_n, + \lambda_n)}(\Delta)$, i.e. the functional calculus on $\Delta$ evaluated in $\chi$, the indicator function. Then, $$ \begin{aligned} \|\Delta P_n f - \Delta\| & = \|\Delta \chi_{(-\lambda_n, + \lambda_n)}(\Delta) f - \Delta f\| \\ & = \|(id \cdot \chi_{(-\lambda_n, + \lambda_n)})(\Delta) f - \Delta f\| \\ & = \|(id \cdot \chi_{(-\lambda_n, + \lambda_n)} - id)(\Delta) f\|. \end{aligned} $$ First question: Is $P_n = \chi_{(-\lambda_n, + \lambda_n)}(\Delta)$? Further, I am a) not sure if this works, b) not sure if the measurable functional calculus is continuous in the unbounded case c) not happy that $\|id \cdot \chi_{(-\lambda_n, + \lambda_n)} - id \|_{L^{\infty}} \rightarrow 0$ does not hold, since the identity is defined on the spectrum of $\Delta$ which is unbounded in general. Can somebody answer some of those question, or has another solution?