How to Find Eigenstates for Value 0 in Projection on Coherent States Paper?

[Heidi]

Homework Statement

this is about coherent states

Relevant Equations

|z><z| = :exp(a-z)^\dagger (a-z):

i am reading this paper . in the definition 19 we have
|z><z| = :exp(a-z)^\dagger (a-z):
in the extansion the first term is the identity son it is not hart to find an eigenvector for the value 1. it is ok if the vector is annihilated by a. if is the case for the coherent grouns state. how to find an eigenstate for the value 0 ?

 

[Heidi]

It would be simpler to begin with z = 0>

 

[Heidi]

I made a typo. The definition is not $ |z><z| = :exp(a-z)^\dagger (a-z): $
it is

$ |z \rangle \langle z| = :exp-(a-z)^\dagger (a-z):$
$