Non-Hermitian Hamiltonian versusE= 0 localized states

20 08 Non - Hermitian Hamiltonian versus E = 0 localized states

We analyze the zero energy solutions, of a two dimensional system which undergoes a non-radial symmetric, complex potential V (r, φ). By virtue of the coherent states concept, the localized states are constructed, and the consequences of the imaginary part of the potential are found both analytically and schematically.

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...

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...

On Scattering from Localized non-Hermitian Potentials

We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square-wells or deltafunctions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processe...

Minimal length in Quantum Mechanics and non-Hermitian Hamiltonian systems