For any weight parameter in the improved-dynamics LQC Hamiltonian, negative weights give essentially self-adjoint operators, while positive weights require U(1)-labeled self-adjoint extensions, which the paper implements in a propagator.
Weinberg, The Cosmological Constant Problem, Rev
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Unitary evolution and cosmic acceleration in Loop Quantum Cosmology
For any weight parameter in the improved-dynamics LQC Hamiltonian, negative weights give essentially self-adjoint operators, while positive weights require U(1)-labeled self-adjoint extensions, which the paper implements in a propagator.