QM I - Decomposition of countable basic states into coherent states

[BasslineSanta]

Homework Statement

Consider a quantum system with a countable number of basic states $\left|n\right\rangle$.
Calculate the decomposition into a basis of coherent states $\left|λ \right\rangle$ all obeying $\hat{a}$ $\left|λ \right\rangle$ = λ $\left|λ \right\rangle$

Homework Equations

$\hat{a}$ is the lowering operator:
$\hat{a} \left|n\right\rangle$ = √n $\left|n-1\right\rangle$

The Attempt at a Solution

Because $\left|λ\right\rangle$ form a basis, i can equate $\left|n\right\rangle$ = Ʃλ$_{n}\left|λ\right\rangle$.
Applying the lowering operator n-times to both sides of the equation, i get: √n! $\left|0\right\rangle$ = λ$^{n}$ Ʃλ$_{n}$ $\left|λ\right\rangle$
By equality of two vectors i can say that √n! = λ$^{n}$ and that $\left|0\right\rangle$ = Ʃλ$_{n}$ $\left|λ\right\rangle$.

Now i got kinda stuck. I thought if i get to the $\left|0\right\rangle$ vector, i can just keep applying the raising operator to get any state $\left|n\right\rangle$ written in my new vectors $\left|λ\right\rangle$. But i realized i do not know how the raising operator acts on them. Neither do i know if i chose the right approach, but it feels like the only thing i could have done, considering the information given.

I would really appreciate some help.
Thanks a lot in advance!

 

[Oxvillian]

I'm a little confused here as to whether you're trying to express a coherent state as a sum of the |n>'s, or one of the |n>'s as a sum of coherent states

The former is easy enough, but the latter is trickier - the answer won't be unique, because the coherent state basis is an overcomplete basis. Probably you'll have to use a coherent state resolution of unity, which involves integrating over the complex λ plane.