6 research outputs found
Exact propagators for SUSY partners
Pairs of SUSY partner Hamiltonians are studied which are interrelated by
usual (linear) or polynomial supersymmetry. Assuming the model of one of the
Hamiltonians as exactly solvable with known propagator, expressions for
propagators of partner models are derived. The corresponding general results
are applied to "a particle in a box", the Harmonic oscillator and a free
particle (i.e. to transparent potentials).Comment: 31 page
Interactions of Hermitian and non-Hermitian Hamiltonians
The coupling of non-Hermitian PT-symmetric Hamiltonians to standard Hermitian
Hamiltonians, each of which individually has a real energy spectrum, is
explored by means of a number of soluble models. It is found that in all cases
the energy remains real for small values of the coupling constant, but becomes
complex if the coupling becomes stronger than some critical value. For a
quadratic non-Hermitian PT-symmetric Hamiltonian coupled to an arbitrary real
Hermitian PT-symmetric Hamiltonian, the reality of the ground-state energy for
small enough coupling constant is established up to second order in
perturbation theory.Comment: 9 pages, 0 figure
The brachistochrone problem in open quantum systems
Recently, the quantum brachistochrone problem is discussed in the literature
by using non-Hermitian Hamilton operators of different type. Here, it is
demonstrated that the passage time is tunable in realistic open quantum systems
due to the biorthogonality of the eigenfunctions of the non-Hermitian Hamilton
operator. As an example, the numerical results obtained by Bulgakov et al. for
the transmission through microwave cavities of different shape are analyzed
from the point of view of the brachistochrone problem. The passage time is
shortened in the crossover from the weak-coupling to the strong-coupling regime
where the resonance states overlap and many branch points (exceptional points)
in the complex plane exist. The effect can {\it not} be described in the
framework of standard quantum mechanics with Hermitian Hamilton operator and
consideration of matrix poles.Comment: 18 page
PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are
studied. For models with constant non-Abelian gauge potentials and extended
parity inversions compact and noncompact Lie group components are analyzed via
Cartan decompositions. A Lie triple structure is found and an interpretation as
PT-symmetrically generalized Jaynes-Cummings model is possible with close
relation to recently studied cavity QED setups with transmon states in
multilevel artificial atoms. For models with Abelian gauge potentials a hidden
Clifford algebra structure is found and used to obtain the fundamental symmetry
of Krein space related J-selfadjoint extensions for PTQM setups with
ultra-localized potentials.Comment: 11 page
The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as
quantum system consisting of a non-Hermitian PTS component and a purely
Hermitian component simultaneously. Interpreting this specific setup as
subsystem of a larger Hermitian system, we find non-unitary operator
equivalence classes (conjugacy classes) as natural ingredient which contain at
least one Dirac-Hermitian representative. With the help of a geometric analysis
the compatibility of the vanishing passage time solution of a PTS
brachistochrone with the Anandan-Aharonov lower bound for passage times of
Hermitian brachistochrones is demonstrated.Comment: 12 pages, 2 figures, strongly extended versio
Projective Hilbert space structures at exceptional points
A non-Hermitian complex symmetric 2x2 matrix toy model is used to study
projective Hilbert space structures in the vicinity of exceptional points
(EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are
Puiseux-expanded in terms of the root vectors at the EP. It is shown that the
apparent contradiction between the two incompatible normalization conditions
with finite and singular behavior in the EP-limit can be resolved by
projectively extending the original Hilbert space. The complementary
normalization conditions correspond then to two different affine charts of this
enlarged projective Hilbert space. Geometric phase and phase jump behavior are
analyzed and the usefulness of the phase rigidity as measure for the distance
to EP configurations is demonstrated. Finally, EP-related aspects of
PT-symmetrically extended Quantum Mechanics are discussed and a conjecture
concerning the quantum brachistochrone problem is formulated.Comment: 20 pages; discussion extended, refs added; bug correcte