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Then, the nonlinear diffeomorphism transformation is introduced to obtain a transformed system with block triangular structure and the interconnections among the subsystems are allowed.
From the inverse transformation of the eigenstates associated to the transformed system, one can evaluate full eigenstates in the original system.
By executing inverse transformation for the wave function obtained in the transformed system, we derived the exact wave function associated to the DSN in the original system.
By inverse transformation of quantum solutions (wave functions) identified in the transformed system, we also obtained the full wave functions for the corresponding superposition states in the original system [see Eq. (38) with Eqs. (39) and (40)].
The techniques presented here reduce the order of this transformed system and all original states are obtained via the appropriate transformations.
The transformed system is proved to be self-adjoint.
This includes control schemes for the transformed system.
Then, we design controllers to stabilize the transformed system.
Next, we analyze the characteristics of the transformed system.
Particular attention has been paid to the boundary conditions of the Fourier transformed system.
An exact closed-form fundamental solution of the transformed system is then derived.
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