Making sense of the divergent series for reconstructing a Hamiltonian from its eigenstates and eigenvalues

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials, this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian