By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results}, author = {Lazar, Martin and Zuazua Iriondo, Enrique}, doi = {10.1016/j.automatica.2024.111653}, faupublication = {yes}, journal = {Automatica}, pages = {111-653}, peerreviewed = {Yes}, title = {{Eigenvalue} bounds for the {Gramian} operator of the heat equation}, url = {https://dcn.nat.fau.eu/wp-content/uploads/2nd{\_}version-gram-lyapunov-autom.pdf}, volume = {164}, year = {2024} }