I have an operator that has a non-trivial solution to the homogeneous equation.
Let $\hat{e}_0$ be the critical eigenvector and C(t) be the corresponding amplitude of an operator $\frac{d}{dt} - \mathcal{L}$ such that $(\frac{d}{dt} - \mathcal{L}) C(t) \hat{e}_0 = 0$.
Otherwise, $\mathcal{L}$ has a discrete spectrum and complete orthonormalised basis of eigenvectors.
I know by the Fredholm alternative, that I cannot now solve this equation uniquely \begin{equation} (\frac{d }{d t} - \mathcal{L}) u = f \end{equation} where $f$ is a forcing term. However, can I still write the solution in the following form?
\begin{equation} u = e^{t\mathcal{L}} u_0 + \int^t_0 e^{\mathcal{L}(t-s)}f(s) ds \end{equation}
How do I specify the homogeneous solution in the above format?
Best wishes,
Catherine