$\mathcal{P}\mathcal{T}$-symmetric Quantum systems for position-dependent effective mass violate the Heisenberg uncertainty principle

We have studied a $\mathcal{P}\mathcal{T}$-symmetric quantum system for a class of position-dependent effective mass. Formalisms of supersymmetric quantum mechanics are utilized to construct the partner potentials. Since the system under consideration is not self-adjoint, the intertwining operators do not factorize the Hamiltonian. We have factorized the Hamiltonian with the aid of generalized annihilation and creation operators, which acts on a deformed coordinate and momentum space. The coherent state structure for the system is constructed from the eigenstates of the generalized annihilation operator. \\ It turns out that the self-adjoint deformed position and momentum operators violate the Heisenberg uncertainty principle for the $\mathcal{P}\mathcal{T}$-symmetric system. This violation depends solely on the $\mathcal{P}\mathcal{T}$-symmetric term, not on the choice of the inner product. For explicit construction, we have demonstrated, for simplicity, a constant mass $\mathcal{P}\mathcal{T}$-symmetric system Harmonic oscillator, which shows the violation of the uncertainty principle for a choice of acceptable parameter values. The result indicates that either $\mathcal{P}\mathcal{T}$-symmetric systems are a trivial extension of usual quantum mechanics or only suitable for open quantum systems.

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