Perturbation and branching of the solutions of nonlinear eigenvalue problems

Riedrich, Thomas

doi:10.1515/9783112541845-042

Cited by 5 publications

(7 citation statements)

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“…Getting back to (57), we see that L pp (A ) = L pp (A). Therefore, A = A , which proves the injectivity of the mapping (26). 19, there exists a unique Z ∈ B η (Z) such that 1 1 (0,1] (γ ) = e Z P 0 e −Z , and by the classical finite-dimensional version of the latter lemma there exists a unique A pf ∈ A pf in the vicinity of 0 such that 1 1 {1} (γ ) = e Z e L pf (0,0,A pf ,0) 1 1 {1} (γ 0 )e −L pf (0,0,A pf ,0) e −Z .…”

Section: Proof Of Lemmamentioning

confidence: 62%

“…Introduced by Rayleigh [22] in the 1870s, it was used for the first time in quantum mechanics in an article by Schrödinger [27] published in 1926. The mathematical study of the perturbation theory of self-adjoint operators was initiated by Rellich [25] in 1937, and has been since then the subject of a large number of contributions in the mathematical literature (see [18,26,29] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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A mathematical perspective on density functional perturbation theory

Cancès

Mourad

2014

Nonlinearity

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This article is concerned with the mathematical analysis of the perturbation method for extended Kohn-Sham models, in which fractional occupation numbers are allowed. All our results are established in the framework of the reduced Hartree-Fock (rHF) model, but our approach can be used to study other kinds of extended Kohn-Sham models, under some assumptions on the mathematical structure of the exchangecorrelation functional. The classical results of Density Functional Perturbation Theory in the non-degenerate case (that is when the Fermi level is not a degenerate eigenvalue of the mean-field Hamiltonian) are formalized, and a proof of Wigner's (2n + 1) rule is provided. We then focus on the situation when the Fermi level is a degenerate eigenvalue of the rHF Hamiltonian, which had not been considered so far.

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Section: Proof Of Lemmamentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

A mathematical perspective on density functional perturbation theory

Cancès

Mourad

2014

Nonlinearity

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“…The parameter dependence of the eigenvalues is, however, extremely complex in general. To obtain ( ) q ¶ ¶ a E k j , we use the perturbation theory of eigenvalue problems (Hellmann-Feynman theorem) 28,29) and we have…”

Section: Fitting Proceduresmentioning

confidence: 99%

“…Here ˆ( ) k b E is a target complex band-structure, which should be calculated with an atomistic model. To use the perturbation theory of eigenvalue problems 28,29) to find a derivative of eigenvalues with respect to q j , we transform the generalized eigenvalue problem of Eq. ( 9) to…”

Section: Fitting Proceduresmentioning

confidence: 99%

Equivalent model for band-to-band tunneling simulation of direct-gap III–V semiconductor nanowires

Okada¹,

Hashimoto²,

Mori³

2021

Jpn. J. Appl. Phys.

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We have proposed an equivalent model for use in the simulation of tunnel devices. The equivalent model reproduces not only the real bandstructure but also the complex band-structure of a target system. It has the features of adjustable spatial resolution and high spatial symmetry. We have tested it by calculating the band-to-band transmission functions and the maximum current of direct-gap III-V semiconductor nanowires, and confirmed that it correctly reproduces the transport characteristics of the target system. The equivalent model enables us to substantially reduce the Hamiltonian matrix size, leading to speed-up the quantum transport simulation of tunnel devices, such as tunnel field-effect transistors.

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“…The spectral theorem for Hermitian matrices is a powerful tool with implications in many areas of mathematics such as combinatorial graph theory [1][2][3], numerical analysis [4][5][6], optimization theory [7], engineering [8][9][10], and most areas of theoretical physics [11]. The wide applicability of the spectral theorem can be mostly attributed to the deep understanding of perturbation theory of eigenvalues and eigenspaces of Hermitian matrices [12,13], and the results are well-known today [14].…”

Section: Introductionmentioning

confidence: 99%

Block perturbation of symplectic matrices in Williamson’s theorem

Babu¹,

K.²

2023

Can. Math. Bull.

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Williamson's theorem states that for any 2𝑛 ×2𝑛 real positive definite matrix 𝐴, there exists a 2𝑛 × 2𝑛 real symplectic matrix 𝑆 such that 𝑆 𝑇 𝐴𝑆 = 𝐷 ⊕ 𝐷, where 𝐷 is an 𝑛 × 𝑛 diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of 𝐴.Let 𝐻 be any 2𝑛 × 2𝑛 real symmetric matrix such that the perturbed matrix 𝐴 + 𝐻 is also positive definite. In this paper, we show that any symplectic matrix S diagonalizing 𝐴 + 𝐻 in Williamson's theorem is of the form S = 𝑆𝑄 + 𝒪(∥𝐻 ∥), where 𝑄 is a 2𝑛 × 2𝑛 real symplectic as well as orthogonal matrix. Moreover, 𝑄 is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of 𝐴. Consequently, we show that S and 𝑆 can be chosen so that ∥ S − 𝑆∥ = 𝒪(∥𝐻 ∥). Our results hold even if 𝐴 has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear

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Perturbation and branching of the solutions of nonlinear eigenvalue problems

Cited by 5 publications

References 0 publications

A mathematical perspective on density functional perturbation theory

A mathematical perspective on density functional perturbation theory

Equivalent model for band-to-band tunneling simulation of direct-gap III–V semiconductor nanowires

Block perturbation of symplectic matrices in Williamson’s theorem

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