I want to find eigenstates of $\hat{x} = a + a^\dagger$ operator, I've started by considering a linear combination like $\sum_{m=0} c_m (a^\dagger)^m |0 \rangle$ and match $c_m$ such that linear combination to be an eigenstate of $\hat{x}$.
But it leads to a recursive relation on $c_m$ which seems hard to evaluate, and I happen to know that eigenstates should be labeled by a complex number $z$, so I don't know how to enter it in my analysis.
Could anyone help me or mention a reference?
The normalized eigenstates of the frequency $\Omega$ Harmonic oscillator are $$ \langle x|n\rangle=\frac{\varphi_n(\sqrt{\Omega_0} x)}{(\Omega_0)^{1/4}} $$ where $$ \varphi_n(x)\equiv \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} H_n(x) e^{-x^2/2}. $$ We want to expand the $\hat x$ eigenvalue $x_0$ eigenstate wavefunction $\langle x|x_0\rangle=\delta(x-x_0)$ in terms of these, but the completeness relation tells us that $$ \sum_{n=0}^\infty \langle x|n\rangle\langle n|x_0\rangle=\delta(x-x_0) $$ and so $$ | x_0\rangle= \sum_n|n\rangle\langle n|x_0\rangle=\sum_n |n\rangle \frac{\varphi_n(\sqrt{\Omega_0} x_0)}{(\Omega_0)^{1/4}} $$
I don't know where complex numbers are coming in. Are you sure that you do not want to introduce coherent states $$ |z\rangle =e^{a^\dagger z}|0\rangle? $$ We do have $$ z\in {\mathbb C}, $$ but these are not eigenstates of $\hat x$, but rather of $\hat a$.
