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Erschienen in:
2019 | OriginalPaper | Buchkapitel
13. Perturbations of Jordan Blocks
verfasst von : Johannes Sjöstrand
Erschienen in: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations
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Abstract
In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A
0, introduced in Sect. 2.4:
$$\displaystyle A_0=\begin {pmatrix}0 &1 &0 &0 &\ldots &0\\ 0 &0 &1 &0 &\ldots &0\\ 0 &0 &0 &1 &\ldots &0\\ . &. &. &. &\ldots &.\\ 0 &0 &0 &0 &\ldots &1\\ 0 &0 &0 &0 &\ldots &0 \end {pmatrix}: {\mathbf {C}}^N\to {\mathbf {C}}^N. $$
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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.
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We have spectral stability (a good resolvent estimate) in \(\mathbf {C}\setminus \overline {D(0,1)}\), since ∥A 0∥ = 1.
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σ(A 0) = {0}.
Thus, if A
δ = A
0 + δQ is a small (random) perturbation of A
0 we expect the eigenvalues to move inside a small neighborhood of \(\overline {D(0,1)}\). In the special case when Qu = (u|e
1)e
N, where \((e_j)_1^N\) is the canonical basis in C
N, we have seen in Sect. 2.4 that the eigenvalues of A
δ are of the form so if we fix 0 < δ ≪ 1 and let N →∞, the spectrum “will converge to a uniform distribution on S
1”.
$$\displaystyle \delta ^{1/N}e^{2\pi ik/N},\ k\in \mathbf {Z}/N\mathbf {Z}, $$