Step 1: Write the heat equation as $u_t=\alpha^2 \nabla^2 u$, where $\nabla^2$ is the Laplacian operator (here $\nabla^2=\partial^2/\partial x^2$ in one dimension).
Step 2: Separation of variables $u=X(x)T(t)$ converts the PDE into the operator eigenvalue equation \[\nabla^2 X=-\lambda X\] together with the given boundary conditions on $X$.
Step 3: This is a Sturm-Liouville eigenvalue problem: $\nabla^2$ (with the stated boundary conditions) is a self-adjoint linear operator, and nontrivial solutions $X_n(x)$ exist only for a discrete set of real, non-negative numbers $\lambda_n$.
Step 4: These $\lambda_n$ are by definition the eigenvalues of the operator, and the associated $X_n(x)$ (for example $\sin(n\pi x/L)$ for Dirichlet conditions) are its eigenfunctions.
Step 5: Because the eigenfunctions of a self-adjoint operator form a complete orthogonal set, they let any initial condition be expanded as $u(x,0)=\sum_n c_nX_n(x)$; the time factor for each mode then decays as $T_n(t)=e^{-\alpha^2\lambda_n t}$. This confirms that the separation constants $\lambda_n$ are exactly the eigenvalues of the Laplacian operator. \[\boxed{\text{The eigenvalues of the Laplacian operator}}\]