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Perturbations of Jordan Blocks

  • Johannes Sjöstrand
Chapter
Part of the Pseudo-Differential Operators book series (PDO, volume 14)

Abstract

In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A0, introduced in Sect.  2.4:
$$\displaystyle A_0=\begin {pmatrix}0 &1 &0 &0 &\ldots &0\\ 0 &0 &1 &0 &\ldots &0\\ 0 &0 &0 &1 &\ldots &0\\ . &. &. &. &\ldots &.\\ 0 &0 &0 &0 &\ldots &1\\ 0 &0 &0 &0 &\ldots &0 \end {pmatrix}: {\mathbf {C}}^N\to {\mathbf {C}}^N. $$
  • Zworski noticed that for every z ∈ D(0, 1), there are associated exponentially accurate quasimodes when N →. Hence the open unit disc is a region of spectral instability.

  • We have spectral stability (a good resolvent estimate) in \(\mathbf {C}\setminus \overline {D(0,1)}\), since ∥A0∥ = 1.

  • σ(A0) = {0}.

Thus, if Aδ = A0 + δQ is a small (random) perturbation of A0 we expect the eigenvalues to move inside a small neighborhood of \(\overline {D(0,1)}\). In the special case when Qu = (u|e1)eN, where \((e_j)_1^N\) is the canonical basis in CN, we have seen in Sect.  2.4 that the eigenvalues of Aδ are of the form
$$\displaystyle \delta ^{1/N}e^{2\pi ik/N},\ k\in \mathbf {Z}/N\mathbf {Z}, $$
so if we fix 0 < δ ≪ 1 and let N →, the spectrum “will converge to a uniform distribution on S1”.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

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