Suppose $\mathcal{L}$ is a bounded linear operator and I have the solution to Eigenvalue problem

$\mathcal{L} \phi + \lambda \phi = 0$

wish to solve the following PDE

$\left(-\partial_t + \mathcal{L}\right)u = 0$.

If the spectrum of $\mathcal{L}$ is continuous or discrete, then a general solution to the PDE is

$\int C_q e^{- \lambda_q t} \phi_q dq$

or

$\sum_q C_q e^{- \lambda_q t} \phi_q$,

where the $C$'s are constants.

But, what if the spectrum of $\mathcal{L}$ is mixed and has a continuous part, a discrete part, and a singular part? Is there a general way to write the solution to the above PDE if I do not know the spectrum of $\mathcal{L}$?

This has come up in my research because I need to work with the $e^{- \lambda_q t}$ but I do not know what the spectrum of $\mathcal{L}$?