spectral projections and resolvent bounds for quantized...
TRANSCRIPT
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Spectral projections and resolvent bounds forquantized partially elliptic quadratic forms
Joe Viola
Laboratoire de Mathématiques Jean LerayUniversité de Nantes
14 March 2013
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Examples and resolvent growth
We will discuss the class of elliptic(ish) quadratic operators Q.As examples, consider the complex harmonic oscillator
Qθ = −e−2iθd2
dx2+ e2iθx2,
studied by E. B. Davies and others, and quadraticKramers-Fokker-Planck
Pa =12
(v2 − ∂2v ) + a(v∂x − x∂v).
Question: What is the behavior of the resolvent norm
||(Q− z)−1||L(L2), |z| → ∞?
Particularly, what is the rate of exponential growth deep withinthe numerical range, close to the spectrum?
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log10 ||(Pa − z)−1||L(L2) for a = 1/√
2
0 5 10 150
2
4
6
8
10
12
14
16
18
0
2
4
6
8
10
12
14
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Spectral projections
We may define spectral projections around an eigenvalue “µα”via
Πα =1
2πi
∫|z−µα|=ε
(z− Q)−1 dz.
If ||Πα|| large, then ||(Q− z)−1|| somewhere large, but not viceversa (cancellation).
We may find vα with ||vα|| = 1 and
||Πα|| = ||Παvα||.
If ||Παvα|| is large, maybe so is ||(Q− z)−1vα||?
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Good approximation by ||(Pa − z)−1vα|| for a = 1/√
2For corresponding α, compute relative error
log10
(||(Pa − z)−1||||(Pa − z)−1vα||
− 1).
10.5 11 11.5
2.8
3
3.2
3.4
3.6
3.8
4
4.2
10.5 11 11.5
2.8
3
3.2
3.4
3.6
3.8
4
4.2
10.5 11 11.5
2.8
3
3.2
3.4
3.6
3.8
4
4.2
10.5 11 11.5
2.8
3
3.2
3.4
3.6
3.8
4
4.2
10.5 11 11.5
2.8
3
3.2
3.4
3.6
3.8
4
4.2
7
6
5
4
3
2
1
0
1
2
3