In this paper, we investigate a class of
PT-symmetric quantum systems with position-dependent effective mass (PDEM). We factorize the PDEM Hamiltonian using a pair of first order differential operators. The resulting commutation relation introduces the notion of a position-dependent effective Planck parameter, which reduces to Planck's constant in the conventional Hermitian quantum system with constant mass. These factorization operators define a deformed phase space, within which we construct generalized coherent states to first order in the
PT-symmetry parameter. We then revisit the Heisenberg uncertainty principle for this state and demonstrate that the deformed position and momentum operators satisfy the corresponding uncertainty relation in the
PT-symmetric PDEM framework. Finally, we present explicit computational results for a toy-model oscillator with position-dependent effective mass in a
PT-symmetric setting. In the conclusions section, we outline a gedankenexperiment for experimental determination of
PT-symmetric parameter.
In the context of the factorization method, we investigate the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass. By considering a specific modification in the superpotential, suitable annihilation and creation operators are constructed in order to reproduce the Hermitian counterpart Hamiltonian in the factorized form. We show that by means of these ladder operators, we can construct a wide range of exactly solvable potentials as well as their accompanying coherent states. Alternatively, we explore the relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterparts, obtained from a similarity transformation, to construct the associated pseudo-Hermitian coherent states. These latter preserve the structure of Perelomov's states and minimize the generalized position–momentum uncertainty principle. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
We propose a mathematically rigorous unified framework for hybrid quantum mechanics that systematically combines algebraic deformation and spatial non-locality within a single operator formalism. By constructing a self-adjoint hybrid kinetic operator through spectral calculus, we derive exact generalized uncertainty relations that interpolate between
q-deformed and fractional quantum mechanical bounds. Furthermore, we establish a rigorous quantum speed limit theorem for the hybrid Hamiltonian, revealing how deformation parameters, fractional orders, and external potentials tune the fundamental evolution rate of quantum states. We prove that algebraic deformation accelerates coherent dynamics through discrete momentum quantization, while fractional non-locality induces spectral broadening that suppresses evolution speed. The framework recovers standard quantum mechanics,
q-quantum mechanics, and fractional quantum mechanics as limiting cases, and provides explicit phenomenological signatures for experimental discrimination in trapped-ion, superconducting, and cold-atom platforms.
The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.