Self-adjoint time operator is the rule for discrete semi-bounded Hamiltonians

We prove explicitly that to every discrete semi-bounded Hamiltonian with constant degeneracy, and with a finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space, there exists a characteristic self-adjoint time operator that is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate to the same Hamiltonian in the same dense subspace.