# Help understanding the so-called "spectral method"

This is a follow-up question to an answer I read here. $$M$$ is some hermitian matrix and $$V$$ an vector.

Since the matrix is hermitian, you could use it as a hamiltonian to propagate it in imaginary time. That is, solve the following system of differential equations:

$$i\frac{d \vec{V}}{dt}=M\vec{V}$$

The general solution to this is:

$$V(t)=V_0e^{iMt}$$

Then you take your $$\vec{V(t)} \cdot \vec{V(0)}$$, fourier transform it, and the height and placement of the peaks will tell you the components along various eigenvectors and their associated eigenvalues. This is sometimes called "the spectral method" in ultrafast atomic physics.

I want to understand this method.

My thoughts on an intuitive level so far: If I take the second equation as given, $$V(t)$$ is a vector-valued process for $$t$$. The scalar product $$\left$$ encodes the matrix $$e^{iMt}$$ in a scalar-valued object. The fourier transform for a scalar $$e^{imt}$$ is the delta distribution $$\delta_{m}$$, up to possibly rescaling by $$2\pi$$. That is, the fourier transform decodes information of test functions $$m\mapsto e^{imt}$$. So the foruier transform of $$\left$$ might somehow decode information about the hermitian matrix $$M$$.

Is this intuition right so far? If so, any help on how to make this rigorous and how to understand what is going on? A small explicit example would be very helpful.

Any source such as a textbook, lecture notes etc. is most welcome.