# Projective Hilbert space structures at exceptional points

@article{Gnther2007ProjectiveHS,
title={Projective Hilbert space structures at exceptional points},
author={Uwe G{\"u}nther and Ingrid Rotter and Boris F. Samsonov},
journal={Journal of Physics A},
year={2007},
volume={40},
pages={8815-8833}
}
• Published 10 April 2007
• Mathematics
• Journal of Physics A
A non-Hermitian complex symmetric 2 × 2-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 Puiseuxexpanded 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 behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The…
86 Citations
Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type
• Mathematics
• 2014
We propose a noncommutative version of the Euclidean Lie algebra E2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra
Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation
• Mathematics, Physics
• 2008
For an arbitrary possibly non-Hermitian matrix Hamiltonian H, that might involve exceptional points, we construct an appropriate parameter space M and the lines bundle L^n over M such that the
Geometry of PT-symmetric quantum mechanics
• Mathematics
• 2007
Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of
{\cal PT} symmetry breaking and exceptional points for a class of inhomogeneous complex potentials
• Mathematics
• 2009
We study a three-parameter family of -symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk–Schultz models. We show that real eigenvalues merge and become complex at quadratic and
A unifying E2-quasi-exactly solvable model
A new non-Hermitian E2-quasi-exactly solvable model is constructed containing two previously known models of this type as limits in one of its three parameters. We identify the optimal finite
Non-unitary operator equivalence classes, the PT-symmetric brachistochrone problem and Lorentz boosts
• Physics
• 2007
The PT −symmetric (PTS) quantum brachistochrone problem is re-analyzed as a composite quantum system consisting of a non-Hermitian PTS component and purely Hermitian component simultaneously. A
PT symmetry on the lattice: the quantum group invariant XXZ spin chain
• Mathematics, Physics
• 2007
We investigate the PT-symmetry of the quantum group invariant XXZ chain. We show that the PT-operator commutes with the quantum group action and also discuss the transformation properties of the
E2-quasi-exact solvability for non-Hermitian models
We propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters.
Exceptional points in atomic spectra and Bose-Einstein condensates
Exceptional points are a special type of degeneracy which can appear for the resonances of parameter-dependent quantum spectra described by non-Hermitian Hamiltonians. They represent positions in the
Exceptional points in open quantum systems
• Physics
• 2008
Open quantum systems are embedded in the continuum of scattering wavefunctions and are naturally described by non-Hermitian Hamilton operators. In the complex energy plane, exceptional points appear

## References

SHOWING 1-10 OF 153 REFERENCES
Geometric phase around exceptional points
• Physics, Mathematics
• 2005
A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory
Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Mathematics
• 1998
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new
Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states
• Physics
• 2006
We consider quantum mechanics with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells, in particular, biorthogonal bases. The 'self-orthogonality'
Complex extension of quantum mechanics.
• Physics
Physical review letters
• 2002
If PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian, and this work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.
Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians
We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal
Unfolding a diabolic point: a generalized crossing scenario
• Physics
• 2003
The typical avoided crossings for Hermitian quantum systems depending on parameters, the diabolic crossing scenario, are generalized to the non-Hermitian case, e.g. for resonances. Two types of
Making sense of non-Hermitian Hamiltonians
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy