Generalized eigenfunctions of the normal operator and resolution of identity

Question

Consider a Hilbert space $\mathcal{H}\simeq L^2(\mathbb{R}^N)\otimes V$ where $V\simeq \mathbb{C}^m$ is a finite-dimensional Hilbert space and an unbounded normal operator $\hat{L}$ densely defined on $\mathcal{H}$. Suppose that we are able to find the general solution of the equation \begin{equation} \hat{L}\psi_\lambda(x,n)=\lambda \psi_\lambda(x,n),\qquad x\in\mathbb{R}^N,\quad n=1,\cdots,m \end{equation} for $\lambda\in D\subset \mathbb{C}$ and able to show that the equation has no solution for $\lambda\not\in D$. For all $\lambda\in D$, $\psi_\lambda\not\in\mathcal{H}$ but they are tempered distributions. Suppose that we are also able to show that $\psi_\lambda$ are normalized in the sense of delta function: \begin{equation} \langle \psi_\lambda\mid\psi_{\lambda'}\rangle=\delta(\lambda,\lambda'). \end{equation} where $\delta(\lambda,\lambda')$ is the delta function on $D$. Based on the above assumptions, can one conclude the following resolution of identity (with certain measure $d\lambda$) on $\mathcal{H}$? \begin{equation} \int_D d\lambda \mid \psi_\lambda\rangle\langle \psi_{\lambda}\mid=\mathrm{Id}_{\mathcal{H}} \end{equation}