For the reflectionless sech-squared potential, the continuum eigenstates integrate to the Dirac delta minus the bound-state projector, explicitly proving completeness when the bound state is included.
Appel, Mathematics for Physics and Physicists , Princeton University Press, Princeton (2007), pp
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Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics
For the reflectionless sech-squared potential, the continuum eigenstates integrate to the Dirac delta minus the bound-state projector, explicitly proving completeness when the bound state is included.