Degeneracy of Resonances: Branch Point and Branch Cuts in Parameter Space

Hernández, Eduardo; Jáuregui, A.; Mondragón, Alfonso; Nellen, L.

doi:10.1007/s10773-006-9325-7

Cited by 8 publications

(4 citation statements)

References 25 publications

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“…In a companion paper (Hernández et al, 2007), we argued that, when discussing the mixing properties of an isolated doublet of unbound states, it is convenient to write the S−matrix poles, k 1 , and k 2 , in terms of a pole position function k 1,2 as…”

Section: Sections Of the Energy Hypersurfacementioning

confidence: 99%

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Crossings and Anticrossings of Unbound States

Hernández¹,

Jáuregui

Mondragón³

2007

Int J Theor Phys

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We investigate the characteristic crossings and anticrossings of energies and widths of a doublet of resonances, observed in the vicinity of, and at a degeneracy of unbound states, when the control parameters of the system are varied. This characteristic behavior is explained in terms of the local, topological structure of the surfaces that represent the complex energy eigenvalues in parameter space in the vicinity of a degeneracy point. In the simple but illustrative case of the scattering of a beam of particles by a double barrier potential well with two regions of trapping, we solved numerically the implicit, transcendental equation that defines the eigenwave numbers of a degenerate isolated doublet of resonances as functions of the real, control parameters of the system. We found that, at a degeneracy of unbound states, the surface representing the resonance eigenwave numbers as functions of the control parameters has an algebraic branch point of rank one. Unfolding the degeneracy point, crossings and anticrossings of energies and widths are obtained as projections of sections of the eigenwave number surfaces.

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Section: Sections Of the Energy Hypersurfacementioning

confidence: 99%

“…In Hernández et al (2007), it was shown that the functions K(d, V 3 ) and k 2 1,2 (d, V 3 ) are regular functions of the parameter (d, V 3 ) at the exceptional point (d * , V * 3 ) and may be expanded in a Taylor series about this point. Hence…”

Section: Sections Of the Energy Hypersurfacementioning

confidence: 99%

Crossings and Anticrossings of Unbound States

Hernández¹,

Jáuregui

Mondragón³

2007

Int J Theor Phys

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“…Some of the earliest examples of EPDs have been also observed in structures with spatial periodicity which are explored, for instance, in [13][14][15][16], such as those exhibiting degenerate band edges or stationary inflection points. Although EPDs are usually viewed from a linear algebra standpoint, and are associated with systems described by matrices with Jordan blocks [1,13], it has been observed that they also represent points in configuration space where multiple branches of spectra connect, and are linked to branch points in the space of control variables [17,18].…”

Section: Introductionmentioning

confidence: 99%

Exceptional Points of Degeneracy and Branch Points for Coupled Transmission Lines—Linear-Algebra and Bifurcation Theory Perspectives

Hanson

Yakovlev

Othman³

et al. 2019

IEEE Trans. Antennas Propagat.

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We demonstrate several new aspects of exceptional points of degeneracy (EPD) pertaining to propagation in two uniform coupled transmission-line structures. We describe an EPD using two different approachesby solving an eigenvalue problem based on the system matrix, and as a singular point from bifurcation theory, and the link between these two disparate viewpoints. Cast as an eigenvalue problem, we show that eigenvalue degeneracies are always coincident with eigenvector degeneracies, so that all eigenvalue degeneracies are implicitly EPDs in two uniform coupled transmission lines. Furthermore, we discuss in some detail the fact that EPDs define branch points (BPs) in the complex-frequency plane; we provide simple formulas for these points, and show that parity-time (PT) symmetry leads to real-valued EPDs occurring on the real-frequency axis. We discuss the connection of the linear algebra approach to previous waveguide analysis based on singular points from bifurcation theory, which provides a complementary viewpoint of EPD phenomena, showing that EPDs are singular points of the dispersion function associated with the fold bifurcation. This provides an important connection of various modal interaction phenomena known in guided-wave structures with recent interesting effects observed in quantum mechanics, photonics, and metamaterials systems described in terms of the EPD formalism.

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“…A detailed account of these and other results will be published elsewhere [22,23] V. SUMMARY AND CONCLUSIONS…”

Section: R E Immentioning

confidence: 87%

Unfolding a degeneracy point of two unbound states: crossings and anticrossings of energies and widths

Hernández¹,

Jáuregui²,

Mondragón³

et al.

Proceedings. 2005 International Conference Physics and Control, 2005.

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We show that when an isolated doublet of unbound states of a physical system becomes degenerate for some values of the control parameters of the system, the energy hypersurfaces representing the complex resonance energy eigenvalues as functions of the control parameters have an algebraic branch point of rank one in parameter space. Associated with this singularity in parameter space, the scattering matrix, S ℓ (E), and the Green's function, G (+) ℓ (k; r, r ′ ), have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of a typical degeneracy point of two unbound states in parameter space by means of a universal 2-parameter family of functions which is contact equivalent to the pole position function of the isolated doublet of resonances at the exceptional point and includes all small perturbations of the degeneracy condition up to contact equivalence.

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Degeneracy of Resonances: Branch Point and Branch Cuts in Parameter Space

Cited by 8 publications

References 25 publications

Crossings and Anticrossings of Unbound States

Crossings and Anticrossings of Unbound States

Exceptional Points of Degeneracy and Branch Points for Coupled Transmission Lines—Linear-Algebra and Bifurcation Theory Perspectives

Unfolding a degeneracy point of two unbound states: crossings and anticrossings of energies and widths

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