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Continuity of Weighted Dirac Spectra

Zixuan Qiu, Ruijun Wu

Abstract

For the weighted Dirac eigenvalue problem, we show that the two-sided weighted spectrum depends continuously on the weight under continuous deformations within a uniformly elliptic class. Moreover, for differentiable families of weights we obtain a quantitative Lipschitz estimate for the full spectrum in the arsinh--metric, based on a weighted Hellmann--Feynman variational identity.

Continuity of Weighted Dirac Spectra

Abstract

For the weighted Dirac eigenvalue problem, we show that the two-sided weighted spectrum depends continuously on the weight under continuous deformations within a uniformly elliptic class. Moreover, for differentiable families of weights we obtain a quantitative Lipschitz estimate for the full spectrum in the arsinh--metric, based on a weighted Hellmann--Feynman variational identity.

Paper Structure

This paper contains 6 sections, 14 theorems, 163 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Encoding of Dirac spectra
  4. Hellmann--Feynman variational identity
  5. The Continuity of Weighted Spectra with C0 Weight
  6. The Lipschitz Continuity of Weighted Spectra with C1 Weight

Key Result

Theorem 1.3

Assuming hyp:H0, the spectral map is continuous. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Three-dimensional branch-labelled view of the local weighted spectrum.
  • Figure 2: Projection onto the $(t,\lambda)$-plane and the apparent kink after sorting.

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: Nowaczyk2013continuity
  • Remark 2.3
  • Lemma 3.1: Rajendra1997AX-XBequalsY
  • Lemma 3.2
  • proof
  • ...and 20 more