I have the Klein-Gordon equation: $$\big{(}\frac{\partial^{2}}{\partial t^{2}}-\Delta+m^{2}\big {)}\phi(t,x) = 0.$$ I am assuming $\phi = \phi(t,x)$ to be defined on $\mathbb{R}\times B$ where $B$ is a box $B = [-L,L]^{d} \subset \mathbb{R}^{d}$. I know from my physics book that $\phi(t,x)$ can be expanded as follows: $$\phi(t,x) = \sum_{p \in B^{*}}\alpha(t)e_{p}(x)$$ where $B^{*}$ is the momentum space associated with $B$ and for each momentum $p \in B^{*}$, $e_{p}(x)$ is an eigenvalue of the Laplacian $\Delta$.
I want to be able to justify the above formulas in a rigorous way. As far as I understand, the momentum space $B^{*}$ is a (countably infinite) lattice and the eigenvectors of the Laplacian are $e_{p}(x) = L^{-d/2}e^{i p \cdot x}$, with $p\cdot x$ being the inner product in $\mathbb{R}^{d}$ and I believe this is a basis for $L^{2}(B)$. However, why can I expand $\phi(t,x)$ as before? That is, (at each term of the sum) why can I separate the time entry and the position entry $\alpha(t)e_{p}(x)$?
Well, the statement that a solution $\phi$, if it exists, admits this kind of decomposition is simply justified by the fact that, as you said, the eigenvectors $\left\{e_p(x)\right\}$ form a basis of $L^2(B)$: we do not even need that $\phi$ is a solution but just that $\phi(t,\cdot) \in L^2(B)$ for any $t \in \mathbb{R}$ (true if $\phi$ is a regular solution, or at least continuous).
However, in (mathematical) physics people are not interested in expanding solutions of PDEs using random bases, but in finding a basis which has a physical meaning. Indeed, the significant aspect is that each single term of the sum $$\phi_p(t,x)=\alpha_p(t)e_p(x)$$ is itself a solution of the original PDE, and so we can expand a generic solution as a sum of "elementary" solutions. This approach allows us to construct by hand, in a very physical manner, a generic solution of the equation (starting from the elementary ones and using the superposition principle, which by the way requires some justifications for infinite sums) rather than relying on abstract existence theorems.
About the existence of separable elementary solutions of this form: as far as I know there are no results which provide it in general settings, but this is a strategy that works for many classical PDEs. One substitutes $\phi(t,x)=\alpha(t)\beta(x)$ in the equation in attempt to find this kind of solutions, which is convenient because one obtains two easier equations for $\alpha$ and $\beta$. If we are lucky, we will find a set of solutions $\left\{\alpha_i(t)\beta_i(x)\right\}_i$ with $\left\{\beta_i\right\}_i$ being a basis of $L^2$. In this particular case we have $$\beta(x)\alpha''(t)-\alpha(t)\Delta\beta(x)+m^2\alpha(t)\beta(x)=0$$ which becomes $$\frac{\alpha''(t)}{\alpha(t)}+m^2=\frac{\Delta\beta(x)}{\beta(x)}=C$$ and so $\beta(x)$ has to be an eigenvector of the Laplacian (with $C \in \mathbb{R}$ usually chosen s.t. $\phi$ satisfies some additional conditions, like boundary conditions).