Time evolution of non-Hermitian Hamiltonian systems

Time evolution of non-Hermitian Hamiltonian systems

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly timedependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator p...

Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψi〉 into a specific final state |ψf 〉 can be made arbitrarily small. Additionally, it has been shown that finding the shortest possible time requires only the solution of the two...

Minimal length in Quantum Mechanics and non-Hermitian Hamiltonian systems

An Equivalent Hermitian Hamiltonian for the non-Hermitian −x4 Potential

The potential V (x) = −x4, which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then PT symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and inte...

Faster than Hermitian Time Evolution

For any pair of quantum states, an initial state |I〉 and a final quantum state |F 〉, in a Hilbert space, there are many Hamiltonians H under which |I〉 evolves into |F 〉. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, is held fixed. We can then determine the Hamiltonian H that satisfies this constraint and achieves the transform...