A numerical and analytical investigation into non-Hermitian Hamiltonians (original) (raw)

On the eigenvalues of some non-Hermitian Hamiltonians with space-time symmetry

Cornell University - arXiv, 2014

We calculate the eigenvalues of some two-dimensional non-Hermitian Hamiltonians by means of a pseudospectral method and straightforward diagonalization of the Hamiltonian matrix in a suitable basis set. Both sets of results agree remarkably well but differ considerably from the eigenvalues obtained some time ago by other authors. In particular, we do not observe the multiple phase transitions claimed to occur in one of the anharmonic oscillators.

Nonclassical States for Non-Hermitian Hamiltonians with the Oscillator Spectrum

Quantum Reports

In this paper, we show that the standard techniques that are utilized to study the classical-like properties of the pure states for Hermitian systems can be adjusted to investigate the classicality of pure states for non-Hermitian systems. The method is applied to the states of complex-valued potentials that are generated by Darboux transformations and can model both non- P T -symmetric and P T -symmetric oscillators exhibiting real spectra.

Beyond conventional factorization: non-Hermitian Hamiltonians with radial oscillator spectrum

Journal of Physics: Conference Series, 2008

The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.

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 time dependent. 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 P T -symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.

Nonautonomous Hamiltonians

Journal of statistical physics, 1998

We present a theory of resonances for a class of nonautonomous Hamiltonians to treat the structural instability of spatially localized and time-periodic solu-tions associated with an unperturbed autonomous Hamiltonian. The mechanism of instability is radiative decay, due to ...

Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation

Laser Physics - LASER PHYS, 2007

We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.

Some applications of non-hermitian operators in quantum mechanics and quantum field theory

The aim of this paper is to show the possible significance, and usefulness, of various nonself-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.

Non-Hermitian quantum-classical dynamics

Quantum systems with a non-conserved probability can be described by means of non-Hermitian Hamiltonians and non-unitary dynamics. In this paper, the case in which the degrees of freedom can be partitioned in two subsets with light and heavy masses is treated. A classical limit over the heavy coordinates is taken in order to embed the non-unitary dynamics of the subsystem in a classical environment. Such a classical environment, in turn, acts as an additional source of dissipation (or noise), beyond that represented by the non-unitary evolution. The non-Hermitian dynamics of a Heisenberg two-spin chain, with the spins independently coupled to harmonic oscillators, is considered in order to illustrate the formalism.

Spatial confinement, non-Hermitian Hamiltonians and related problems

The European Physical Journal D, 2021

We treat simple examples of systems described by non-relativistic model Hamiltonians which are unconventional. They are not necessarily Hermitian operators. In practice, they often contain applied external fields which are not necessarily small, and they seek to describe the effects of spatial confinement more realistically than most of the classic calculations (some of them by the present authors). A much studied model of a free particle confined by a non-real potential can be accommodated within the same theoretical framework. Numerical treatment of these and similar problems seems well within the capacity of very modest computer systems, as exemplified by a few tabulations.