Manuscript I-B-VP@3 · I-B-VP-v3.tex · proposition · 93c04eb4

Proposition: Heat kernel and large-time estimate

Friedrichs heat kernel and the massless state · lines 530–572 · proof included or linked

I-B-VP@3#prop:heat, I-B-VP@3#eq:heat

Relative to $d V$ , the Friedrichs heat kernel is

K_{β} (s; x, x^{'}) = \frac{e ^{- Δ z^{2} / (4 s)}}{4 π s} \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{4 π α s} n \in Z \sum e^{i q_{n} (φ - φ^{'})} I_{ν_{n}} (\frac{r r ^{'}}{2 s}) .

For each compact regular spatial set $K$ ,

x, x^{'} \in K sup ∣ K_{β} (s; x, x^{'}) ∣ \leq C_{K} s^{- 3/2 - ν_{*}}, s \geq 1, ν_{*} = \frac{min ( β , 1 - β )}{α} .

At $β = 0$ , $ν_{*} = 0$ .

Proof. The Friedrichs radial transform uses $J_{ν} (k r)$ with measure $k d k$ . Weber's integral is

\int_{0}^{\infty} e^{- s k^{2}} J_{ν} (k r) J_{ν} (k r^{'}) k d k = \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{2 s} I_{ν} (\frac{r r ^{'}}{2 s}) .

Combining the normalized angular modes with the longitudinal heat kernel gives (eq:heat) Watson,DLMF.

On $K$ , the power series gives, uniformly for $s \geq 1$ ,

I_{ν} (\frac{r r ^{'}}{2 s}) \leq C_{K} \frac{( C _{K} / s ) ^{ν}}{Γ ( 1 + ν )} .

The orders grow linearly in $∣ n ∣$ , so their sum is bounded by $C_{K} s^{- ν_{*}}$ . The remaining prefactor is $O (s^{- 3/2})$ .

Displayed formulas in this passage

I-B-VP@3#eq:heat · line 533
$K_{β} (s; x, x^{'}) = \frac{e ^{- Δ z^{2} / (4 s)}}{4 π s} \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{4 π α s} n \in Z \sum e^{i q_{n} (φ - φ^{'})} I_{ν_{n}} (\frac{r r ^{'}}{2 s}) .$
unlabeled · line 544
$x, x^{'} \in K sup ∣ K_{β} (s; x, x^{'}) ∣ \leq C_{K} s^{- 3/2 - ν_{*}}, s \geq 1, ν_{*} = \frac{min ( β , 1 - β )}{α} .$
unlabeled · line 556
$\int_{0}^{\infty} e^{- s k^{2}} J_{ν} (k r) J_{ν} (k r^{'}) k d k = \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{2 s} I_{ν} (\frac{r r ^{'}}{2 s}) .$
unlabeled · line 566
$I_{ν} (\frac{r r ^{'}}{2 s}) \leq C_{K} \frac{( C _{K} / s ) ^{ν}}{Γ ( 1 + ν )} .$