Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

Mailybaev, Alexei A.

doi:10.1002/nla.471

Cited by 22 publications

(41 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These triple-degeneracies EP3 occur twice, and have a cusp-like behaviour, emerging from the EP2-curves, identifiable as an elliptic umbilic catastrophe [56]. This topology is also consistent with the analysis of non Hermitian degeneracies of a two-parameters family of 3×3 matrices, done by Mailybaev [57]. In very strong driving fields the matrix M will loose symmetry [58,59] maintaining the cusps but skewing the topology.…”

Section: Appendix C Eigenvalues Of the Matrix Msupporting

confidence: 83%

Exceptional points for parameter estimation in open quantum systems: analysis of the Bloch equations

2015

View full text Add to dashboard Cite

We suggest to employ the dissipative nature of open quantum systems for the purpose of parameter estimation: the dynamics of open quantum systems is typically described by a quantum dynamical semigroup generator . The eigenvalues of  are complex, reflecting unitary as well as dissipative dynamics. For certain values of parameters defining , non-Hermitian degeneracies emerge, i.e. exceptional points (EP). The dynamical signature of these EPs corresponds to a unique time evolution. This unique feature can be employed experimentally to locate the EPs and thereby to determine the intrinsic system parameters with a high accuracy. This way we turn the disadvantage of the dissipation into an advantage. We demonstrate this method in the open system dynamics of a two-level system described by the Bloch equation, which has become the paradigm of diverse fields in physics, from NMR to quantum information and elementary particles.

show abstract

Section: Appendix C Eigenvalues Of the Matrix Msupporting

confidence: 83%

Exceptional points for parameter estimation in open quantum systems: analysis of the Bloch equations

2015

View full text Add to dashboard Cite

show abstract

“…has an eigenvalue multiplicity, using a modified Newton's method [46]. To proceed with numerical computations, we consider by way of example a middle slab made of PMMA, which is bonded between two steel slabs, whose properties are ρ a = 1200 kg m −3 , µ a = 1.21 GPa, l = 1 cm, ρ b = 7800 kg m −3 , µ b = 78.85 GPa, L = 3 cm.…”

Section: Nhqm Perturbation Theory For Elastodynamics: the Model Pmentioning

confidence: 99%

“…To show that this product indeed delivers such a set, consider first the third term in Eq. (46). Since the scaled function vanishes at infinity, the integral is zero at its upper limit, and we are left with its value at x = l, such that ∞ l ρ b C 2 n+ e 2ik bn (x−l)e iθ e iθ dx = iρ b 4k bn (1 ± cos 2k an l) , (47) where we used that fact that C n+ = cos k an l for the even modes (with a plus sign inside the brackets), and C n+ = sin k an l for the odd modes (minus sign).…”

Section: Theory For Real Perturbations In Non-hermitian Elastodymentioning

confidence: 99%

Linking Scalar Elastodynamics and Non-Hermitian Quantum Mechanics

Gal¹,

Moiseyev²

2020

Phys. Rev. Applied

View full text Add to dashboard Cite

Recent years have seen a fascinating pollination of ideas from quantum theories to elastodynamics-a theory that phenomenologically describes the time-dependent macroscopic response of materials.Here, we open route to transfer additional tools from non-Hermitian quantum mechanics. We begin by identifying the differences and similarities between the one-dimensional elastodynamics equation and the time-independent Schrödinger equation, and finding the condition under which the two are equivalent. Subsequently, we demonstrate the application of the non-Hermitian perturbation theory to determine the response of elastic systems; calculation of leaky modes and energy decay rate in heterogenous solids with open boundaries using a quantum mechanics approach; and construction of degeneracies in the spectrum of these assemblies. The latter result is of technological importance, as it introduces an approach to harness extraordinary wave phenomena associated with non-Hermitian degeneracies for practical devices, by designing them in simple elastic systems. As an example of such application, we demonstrate how an assembly of elastic slabs that is designed with two degenerate shear states according to our scheme, can be used for mass sensing with enhanced sensitivity by exploiting the unique topology near the exceptional point of degeneracy. arXiv:1910.07306v2 [cond-mat.soft]

show abstract

“…In a (more) general setting such configurations correspond to codimension-three varieties [18,20]. Apart from these generic real-to-complex transitions on codimension-one varieties, there may occur higher-order intersections of more than two Riemann sheets simultaneously -on varieties Υ of higher codimension, codim(Υ ) ≥ 2, and with larger Jordan blocks in the spectral decomposition [9,21,22].…”

Section: Krein Space Related Physical Setups and Spectral Phase Transmentioning

confidence: 99%

Third-order spectral branch points in Krein space related setups: $$\mathcal{P}\mathcal{T}$$ -symmetric matrix toy model, MHD α 2-dynamo and extended Squire equation

Günther

Stefani

2005

Czech J Phys

View full text Add to dashboard Cite

The spectra of self-adjoint operators in Krein spaces are known to possess real sectors as well as sectors of pair-wise complex conjugate eigenvalues. Transitions from one spectral sector to the other are a rather generic feature and they usually occur at exceptional points of square root branching type. For certain parameter configurations two or more such exceptional points may happen to coalesce and to form a higher-order branch point. We study the coalescence of two square root branch points semi-analytically for a PTsymmetric 4×4 matrix toy model and illustrate its occurrence numerically in the spectrum of the 2 × 2 operator matrix of the magneto-hydrodynamic α 2 -dynamo and an extended version of the hydrodynamic Squire equation. Krein space related physical setups and spectral phase transitionsSome basic spectral properties of the Hamiltonians of PT -symmetric Quantum Mechanics (PTSQM) [1] can be easily explained from the fact [2, 3] that these Hamiltonians are self-adjoint operators in Krein spaces [4, 5] -Hilbert spaces with an indefinite metric structure. In contrast to the purely real spectra of selfadjoint operators in "usual" Hilbert spaces (with positive definite metric structure), the spectrum of self-adjoint operators in Krein spaces splits into real sectors and sectors with pair-wise complex conjugate eigenvalues. In physical terms these two types of sectors are equivalent to phases of exact PT -symmetry and spontaneously broken PT -symmetry [1]. The reality of the spectrum of a PTSQM Hamiltonian, e.g., with complex potential ix 3 , means that this spectrum is located solely in a real sector and that the operator is quasi-Hermitian 1 ) in the sense of Ref. [7]. In * ) Presented at the 3rd International Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics", Istanbul, Turkey, June 20-22, 2005. 1 ) Because quasi-Hermitian operators form only a restricted subclass of self-adjoint operators in Krein spaces (pseudo-Hermitian operators in the sense of Ref.[6]), the question for the reality of the spectrum seems up to now to be only partially solved. Apart from the requirement for existing PT -symmetry of the differential expression of the operator, a subtle interplay between the operator domain and some, in general not yet sufficiently clearly identified, additional structural aspects of its differential expression seems to be responsible for the quasi-Hermiticity of the operator (exact PT -symmetry of the corresponding PTSQM setup).

show abstract

Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

Cited by 22 publications

References 38 publications

Exceptional points for parameter estimation in open quantum systems: analysis of the Bloch equations

Exceptional points for parameter estimation in open quantum systems: analysis of the Bloch equations

Linking Scalar Elastodynamics and Non-Hermitian Quantum Mechanics

Third-order spectral branch points in Krein space related setups: $$\mathcal{P}\mathcal{T}$$ -symmetric matrix toy model, MHD α 2-dynamo and extended Squire equation

Contact Info

Product

Resources

About