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 $\mathcal{L}$?



question about mixed spectrum of a linear operator $\mathcal{L}$
Well, I have to confess that I am not sure about your goals. In which space are you working? But there is always a way to represent the solution as a generazied exponential function, called operator semigroup. If your operator $\mathcal L$ is indeed bounded, then you can represent the solution via Dunford-Riesz calculus. A good reference for this and much more general cases is Engel-Nagel: A short course on operator semigroups, Springer.


