We extend the definition of eta-weak-pseudo-Hermiticity to the class of potentials endowed with position-dependent mass. The construction of non-Hermitian Hamiltonians through some generating function are obtained. Special cases of potentials are thus deduced.
We investigate in this paper the concept of complementarity, introduced by Bagchi and Quesne [Phys. Lett. A 301, 173 (2002)], between pseudo-Hermiticity and weak pseudo-Hermiticity in a rigorous mathematical viewpoint of coordinate transformations when a system has a position-dependent mass. We first determine, under the modified-momentum, the generating functions identifying the complexified potentials V±(x) under both concepts of pseudo-Hermiticity η̃+ (respectively, weak pseudo-Hermiticity η̃−). We show that the concept of complementarity can be understood and interpreted as a coordinate transformation through their respective generating functions. As a consequence, a similarity transformation that implements coordinate transformations is obtained. We show that the similarity transformation is set up as a fundamental relationship connecting both η̃+ and η̃−. A special factorization η+=η−†η− is discussed in the constant mass case, and some Bäcklund transformations are derived.
Non-Hermitian von Roos Hamiltonian's η-weak-pseudo-Hermiticity and exact solvability
A complexi ed von Roos Hamiltonian is considered and a Hermitian rst-order intertwining di¤erential operator is used to obtain the related position dependent mass -weak-pseudo-Hermitian Hamiltonians. Two user -friendly reference-target maps are introduced to serve for exactsolvability of some non-Hermitian -weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -symmetric Scarf II and a non-Hermitian periodic-type PT -symmetric Samsonov-Roy potentials are used as reference models in a user-friendly reference-target map and the corresponding isospectral Hamiltonians are obtained. It is observed that for each exactly-solvable reference Hamiltonian there is a corresponding set of exactly-solvable target Hamiltonians. PACS numbers: 03.65.Ge, 03.65.Fd, 03.65.Ca
In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.