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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
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?
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
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||?
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