Coherent states of Quantum harmonic oscillator .
The Hamiltonian of Quantum harmonic oscillator is $H=(a^+ a+\frac{1}{2})\hbar \omega$,$a=\sqrt{\frac{m \omega}{2 \hbar}}(\hat{x}+\frac{i \hat{p}}{m \omega})$,$\hat{N}=a^+ a$
The Coherent states are defined as eigenstates of $a$,we mark it $$a|\lambda \rangle=\lambda|\lambda \rangle$$ In $N$-representation,we can show that $$|\lambda \rangle=\sum_n c_n |n\rangle ,c_n=\frac{\lambda ^n}{\sqrt{n!}}e^{-\frac{|\lambda|^2}{2}}$$
my question:
can we give the exact value of $\lambda$?
In N-representation,the matrix representation of $a $ is \begin{equation} \left( \begin{matrix} 0&\sqrt{1}&0&0&0&\cdots\\ 0& 0&\sqrt{2}&0&0&\cdots \\ 0& 0&0&\sqrt{3}&0&\cdots \\ 0& 0&0&0&\sqrt{4}&\cdots \\ 0& 0&0&0&0&... \\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{matrix} \right) \end{equation} I want to calculate the eigenvalues of it. But all eigenvalues are $0$. Is it the reason that on the finite dimension?