Accidental degeneracy and berry phase of resonant states

Mondragón, Alfonso; Hernández, E.

doi:10.1007/bfb0106786

Cited by 10 publications

(8 citation statements)

References 27 publications

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“…In [27] two-level coalescences have been associated with chiral system behavior. The geometric phase at EPs has been discussed in [14,26,27,28,29,30,31,32].…”

Section: Introductionmentioning

confidence: 99%

Projective Hilbert space structures at exceptional points

Günther¹,

Rotter²,

Samsonov³

2007

J. Phys. A: Math. Theor.

121

186

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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 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.

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“…In [27] two-level coalescences have been associated with chiral system behavior. The geometric phase at EPs has been discussed in [14,26,27,28,29,30,31,32].…”

Section: Introductionmentioning

confidence: 99%

Projective Hilbert space structures at exceptional points

Günther¹,

Rotter²,

Samsonov³

2007

J. Phys. A: Math. Theor.

121

186

View full text Add to dashboard Cite

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“…The bound state eigenfunctions u sℓ (k s , r) are also solutions of (1) which satisfy the boundary conditions (14) and (15), but, in this case, k s = iκ s with, κ s > 0, which means that asymptotically the outgoing wave of imaginary argument, f ℓ (−k s , r), decreases exponentially with r and the energy eigenvalue E s is real and negative.…”

Section: Regular and Physical Solutions Of The Radial Equationmentioning

confidence: 99%

Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states

2003

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An accidental degeneracy of resonances gives rise to a double pole in the scattering matrix, a double zero in the Jost function and a Jordan chain of length two of generalized Gamow-Jordan eigenfunctions of the radial Schrödinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in bound and resonant energy eigenfunctions plus a continuum of scattering wave functions of complex wave number. In this biorthonormal basis, any operator f (H (ℓ) r ) which is a regular function of the Hamiltonian is represented by a complex matrix wich is diagonal except for a Jordan block of rank two. The occurrence of a double pole in the Green's function, as well as the non-exponential time evolution of the Gamow-Jordan generalized eigenfunctions are associated to the Jordan block in the complex energy representation.

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“…The problem of the characterization of the singularities of the energy surfaces at a degeneracy of resonances arises naturally in connection with the topological phase of unbound states which was predicted by Hernández, Jáuregui, and Mondragón (1992); Hernández (1996, 1998), and, later and independently, by W. D. Heiss (1999), and which was recently measured by the Darmstadt group (Dembowski et al, 2001(Dembowski et al, , 2003.…”

Section: Introductionmentioning

confidence: 97%

“…Korsch and Mossman (2003) made a detailed investigation of degeneracies of resonances in a symmetric double δ−well in a constat Stark field. Keck et al (2003) extended and generalized the discussion of the Berry phase of resonance states, from the case of unbound states of a hermitian Hamiltonian given in Hernández et al (1992); Hernández (1996, 1998), to the case of unbound states of non-hermitian Hamiltonians.…”

Section: Introductionmentioning

confidence: 97%

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

et al. 2007

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The rich phenomenology of crossings and anticrossings of energies and widths, observed in an isolated doublet of resonances when one control parameter is varied, is fully explained in terms of the topological properties of the energy hypersurfaces close to the degeneracy point. The hypersurface representing the complex resonance eigenvalues, as functions of the control parameters, has an algebraic branch point of rank one, and branch cuts in its real and imaginary parts, in parameter space. Associated with this singularity in parameter space, the scattering matrix, S (E), and the Green's function, G (+) (k; r, r ), have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of any 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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Accidental degeneracy and berry phase of resonant states

Cited by 10 publications

References 27 publications

Projective Hilbert space structures at exceptional points

Projective Hilbert space structures at exceptional points

Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states

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

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