4,451 research outputs found
Exceptional points and double poles of the S matrix
Exceptional points and double poles of the S matrix are both characterized by
the coalescence of a pair of eigenvalues. In the first case, the coalescence
causes a defect of the Hilbert space. In the second case, this is not so as
shown in prevoius papers. Mathematically, the reason for this difference is the
bi-orthogonality of the eigenfunctions of a non-Hermitian operator that is
ignored in the first case. The consequences for the topological structure of
the Hilbert space are studied and compared with existing experimental data.Comment: 9 pages, no figure
Correlations in quantum systems and branch points in the complex plane
Branch points in the complex plane are responsible for avoided level
crossings in closed and open quantum systems. They create not only an exchange
of the wave functions but also a mixing of the states of a quantum system at
high level density. The influence of branch points in the complex plane on the
low-lying states of the system is small.Comment: 10 pages, 2 figure
Clustering of exceptional points and dynamical phase transitions
The eigenvalues of a non-Hermitian Hamilton operator are complex and provide
not only the energies but also the lifetimes of the states of the system. They
show a non-analytical behavior at singular (exceptional) points (EPs). The
eigenfunctions are biorthogonal, in contrast to the orthogonal eigenfunctions
of a Hermitian operator. A quantitative measure for the ratio between
biorthogonality and orthogonality is the phase rigidity of the wavefunctions.
At and near an EP, the phase rigidity takes its minimum value. The lifetimes of
two nearby eigenstates of a quantum system bifurcate under the influence of an
EP. When the parameters are tuned to the point of maximum width bifurcation,
the phase rigidity suddenly increases up to its maximum value. This means that
the eigenfunctions become almost orthogonal at this point. This unexpected
result is very robust as shown by numerical results for different classes of
systems. Physically, it causes an irreversible stabilization of the system by
creating local structures that can be described well by a Hermitian Hamilton
operator. Interesting non-trivial features of open quantum systems appear in
the parameter range in which a clustering of EPs causes a dynamical phase
transition.Comment: A few improvements; 2 references added; 28 pages; 7 figure
A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
We consider non-Hermitian but PT-symmetric extensions of Calogero models,
which have been proposed by Basu-Mallick and Kundu for two types of Lie
algebras. We address the question of whether these extensions are meaningful
for all remaining Lie algebras (Coxeter groups) and if in addition one may
extend the models beyond the rational case to trigonometric, hyperbolic and
elliptic models. We find that all these new models remain integrable, albeit
for the non-rational potentials one requires additional terms in the extension
in order to compensate for the breaking of integrability.Comment: 10 pages, Late
Influence of branch points in the complex plane on the transmission through double quantum dots
We consider single-channel transmission through a double quantum dot system
consisting of two single dots that are connected by a wire and coupled each to
one lead. The system is described in the framework of the S-matrix theory by
using the effective Hamiltonian of the open quantum system. It consists of the
Hamiltonian of the closed system (without attached leads) and a term that
accounts for the coupling of the states via the continuum of propagating modes
in the leads. This model allows to study the physical meaning of branch points
in the complex plane. They are points of coalesced eigenvalues and separate the
two scenarios with avoided level crossings and without any crossings in the
complex plane. They influence strongly the features of transmission through
double quantum dots.Comment: 30 pages, 14 figure
A generalized least-squares framework for rare-variant analysis in family data.
Rare variants may, in part, explain some of the hereditability missing in current genome-wide association studies. Many gene-based rare-variant analysis approaches proposed in recent years are aimed at population-based samples, although analysis strategies for family-based samples are clearly warranted since the family-based design has the potential to enhance our ability to enrich for rare causal variants. We have recently developed the generalized least squares, sequence kernel association test, or GLS-SKAT, approach for the rare-variant analyses in family samples, in which the kinship matrix that was computed from the high dimension genetic data was used to decorrelate the family structure. We then applied the SKAT-O approach for gene-/region-based inference in the decorrelated data. In this study, we applied this GLS-SKAT method to the systolic blood pressure data in the simulated family sample distributed by the Genetic Analysis Workshop 18. We compared the GLS-SKAT approach to the rare-variant analysis approach implemented in family-based association test-v1 and demonstrated that the GLS-SKAT approach provides superior power and good control of type I error rate
Whispering gallery modes in open quantum billiards
The poles of the S-matrix and the wave functions of open 2D quantum billiards
with convex boundary of different shape are calculated by the method of complex
scaling. Two leads are attached to the cavities. The conductance of the
cavities is calculated at energies with one, two and three open channels in
each lead. Bands of overlapping resonance states appear which are localized
along the convex boundary of the cavities and contribute coherently to the
conductance. These bands correspond to the whispering gallery modes appearing
in the classical calculations.Comment: 9 pages, 3 figures in jpg and gif forma
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