Manuscript Hawk02-C@2 · Hawk02-C.tex · section · bb332d66

Semicircle form factor and spectral measure

A regular positive-frequency decay regime › Semicircle form factor and spectral measure · lines 1076–1104

Hawk02-C@2#eq:semicircle, Hawk02-C@2#eq:semig

Fix $c > 0$ and $ω_{0} \geq 3 c$ . Define

w_{c} (x) = ⎩ ⎨ ⎧ \frac{4 c ^{2} - x ^{2}}{2 π c ^{2}}, 0, ∣ x ∣ \leq 2 c, ∣ x ∣ > 2 c,

and

g (ν) = c w_{c} (ν - ω_{0}) .

Its support is the positive band $I_{c} = [ω_{0} - 2 c, ω_{0} + 2 c]$ . This is a regular $L^{2}$ coupling, not a flat distributional coupling. No differentiability at the band edges is assumed.

We have

∥ g ∥_{2} = c, κ_{g} \leq \frac{c ^{2}}{ω _{0} - 2 c} \leq c < ω_{0} .

The full Hamiltonian is therefore nonnegative.

Displayed formulas in this passage

Hawk02-C@2#eq:semicircle · line 1080
$w_{c} (x) = ⎩ ⎨ ⎧ \frac{4 c ^{2} - x ^{2}}{2 π c ^{2}}, 0, ∣ x ∣ \leq 2 c, ∣ x ∣ > 2 c,$
Hawk02-C@2#eq:semig · line 1089
$g (ν) = c w_{c} (ν - ω_{0}) .$
unlabeled · line 1099
$∥ g ∥_{2} = c, κ_{g} \leq \frac{c ^{2}}{ω _{0} - 2 c} \leq c < ω_{0} .$

Semicircle form factor and spectral measure

Displayed formulas in this passage

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