Manuscript En01-B@2 · En01-B-v2.tex · proposition · f226d1eb

Proposition: Quadratic coefficient and remainder

Declared regularized model and its dynamics › Different mass and packet prescriptions · lines 1147–1181 · proof included or linked

En01-B@2#prop:quadratic

Back to manuscript

Define

A_{r_{c}} = \frac{1}{2} \int k_{e}^{2} Q_{r_{c}} (d k), D_{r_{c}}^{(4)} = \frac{1}{24} \int k_{e}^{4} Q_{r_{c}} (d k) .

For $r_{c} > 0$ , these are finite and

0 \leq A_{r_{c}} d^{2} - Γ_{r_{c}} (d) \leq D_{r_{c}}^{(4)} d^{4} .

The quadratic generator is

G_{r_{c}}^{(2)} = - A_{r_{c}} [x, [x, \cdot]] .

For isotropic profiles,

A_{r_{c}} = \frac{G}{3 π ℏ} \int_{0}^{\infty} k^{2} ∣ f_{r_{c}} (k) ∣^{2} d k, D_{r_{c}}^{(4)} = \frac{G}{60 π ℏ} \int_{0}^{\infty} k^{4} ∣ f_{r_{c}} (k) ∣^{2} d k .

Proof. Integrate $0 \leq z^{2} /2 - (1 - cos z) \leq z^{4} /24$ . Angular means of $k_{e}^{2}$ and $k_{e}^{4}$ are $k^{2} /3$ and $k^{4} /5$ . Gaussian decay controls all moments.

Displayed formulas in this passage

unlabeled · line 1150
$A_{r_{c}} = \frac{1}{2} \int k_{e}^{2} Q_{r_{c}} (d k), D_{r_{c}}^{(4)} = \frac{1}{24} \int k_{e}^{4} Q_{r_{c}} (d k) .$
unlabeled · line 1156
$0 \leq A_{r_{c}} d^{2} - Γ_{r_{c}} (d) \leq D_{r_{c}}^{(4)} d^{4} .$
unlabeled · line 1161
$G_{r_{c}}^{(2)} = - A_{r_{c}} [x, [x, \cdot]] .$
unlabeled · line 1165
$A_{r_{c}} = \frac{G}{3 π ℏ} \int_{0}^{\infty} k^{2} ∣ f_{r_{c}} (k) ∣^{2} d k, D_{r_{c}}^{(4)} = \frac{G}{60 π ℏ} \int_{0}^{\infty} k^{4} ∣ f_{r_{c}} (k) ∣^{2} d k .$

Nearby in this section

section: Different mass and packet prescriptions