Manuscript I-B-G@2 · I-B-G-v2.tex · theorem · 3b4d700f

Theorem: Heat, resolvent, and wave representation

The Sommerfeld–Carslaw–Dowker kernel › A precise class of kernel identities · lines 1102–1309 · proof included or linked

I-B-G@2#thm:closed, I-B-G@2#eq:modes, I-B-G@2#eq:flat-massive, I-B-G@2#eq:wave-master, I-B-G@2#eq:massless, I-B-G@2#eq:Sommerfeld

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The following identities hold.

The massless spatial heat kernel is
$H_{β} (τ; x, x^{'}) = \frac{e ^{- Δ z^{2} / (4 τ)}}{4 π τ} \frac{e ^{- (r^{2} + r^{'2}) / (4 τ)}}{4 π α τ} \times n \in Z \sum e^{i (n + β) (φ - φ^{'})} I_{ν_{n}} (\frac{r r ^{'}}{2 τ}), τ > 0.$
At generic pairs, the heat kernel of $B$ equals \[ S_[ (4)^-3/2e^{-m^2-R^2/(4)} ].
For $s > 0$ , the kernel of $(B + s^{2})^{- 1}$ is \[ S_[ {e^{-^2+m^2\,R}}{4 R} ].
Let $F_{m}^{+} (t, R)$ be the flat retarded kernel of $\partial_{t}^{2} - Δ_{R^{3}} + m^{2}$ . For $R > 0$ ,
$F_{m}^{+} (t, R) = \frac{δ ( t - R )}{4 π R} - θ (t - R) \frac{m J _{1} ( m t ^{2} - R ^{2} )}{4 π t ^{2} - R ^{2}},$
where the second term is zero at $m = 0$ , and its quotient has its continuous value at the front. The intrinsic retarded kernel is
$G_{+} (t; x, x^{'}) = S_{β} [F_{m}^{+} (t, \cdot)] .$
For $m = 0$ , at a generic pair,
$G_{+} (t; x, x^{'}) = k \in K (ϑ) \sum e^{2 π i β k} \frac{δ ( t - R _{k} )}{4 π R _{k}} - θ (t - ρ_{0}) \frac{Φ _{β} ( u _{0} , ϑ )}{4 π ^{2} α r r ^{'} sinh u _{0}},$
with
$cosh u_{0} = \frac{t ^{2} - r ^{2} - r ^{'2} - Δ z ^{2}}{2 r r ^{'}} .$
The last term means its locally integrable one-sided extension at $t = ρ_{0}$ .
The formulas are globally bundle-valued distribution identities with the joint-integration prescription in the proof. No arbitrary pointwise value at an image boundary is required. The advanced kernel is
$G_{-} (t; x, x^{'}) = G_{+} (- t; x, x^{'}) .$

Proof. Step 1: the mode heat kernel. The Friedrichs radial operator is diagonalised by the Bessel transform with kernel $J_{ν} (k r)$ . Weber's exponential integral gives

\int_{0}^{\infty} e^{- τ k^{2}} J_{ν} (k r) J_{ν} (k r^{'}) k d k = \frac{1}{2 τ} e^{- (r^{2} + r^{'2}) / (4 τ)} I_{ν} (\frac{r r ^{'}}{2 τ}) .

Together with the angular normalization and the one-dimensional longitudinal heat kernel, this proves (eq:modes). We use the classical integral in Watson; equivalent Bessel identities are recorded in DLMF.

Step 2: the angular summation. For $ζ > 0$ , set

S (ζ, ϑ) = n \sum I_{ν_{n}} (ζ) e^{i (n + β) ϑ / α} .

Use the integral representation

I_{ν} (ζ) = \frac{1}{π} \int_{0}^{π} e^{ζ c o s v} cos (ν v) d v - \frac{sin ( π ν )}{π} \int_{0}^{\infty} e^{- ζ c o s h u - ν u} d u,

valid for the present parameters Watson,DLMF. The first integral is the $n$ -th twisted Fourier coefficient of the piecewise smooth function

h_{ζ} (ϑ) = α k \in Z \sum e^{2 π i β k} e^{ζ c o s (ϑ - 2 π α k)} 1_{{∣ ϑ - 2 π α k ∣ < π}} .

Indeed, integrating over one angular period and changing variables in each summand gives the coefficient

\frac{1}{2 π} \int_{- π}^{π} e^{ζ c o s v} e^{- i (n + β) v / α} d v .

Its imaginary part vanishes, and its real part is the first Bessel integral.

Insert the Abel factor $e^{- ϵ ν_{n}}$ . The first Fourier series tends to $h_{ζ}$ at every generic angle and in local angular $L^{1}$ . The second becomes

- \frac{1}{π} \int_{0}^{\infty} e^{- ζ c o s h u} Φ_{β} (u + ϵ, ϑ) d u .

Lemma lem:Phi supplies local joint integrability, uniformly for small $ϵ$ ; at infinity the factor $e^{- ζ c o s h u}$ dominates. Hence

S (ζ, ϑ) = α k \in K (ϑ) \sum e^{2 π i β k} e^{ζ c o s (ϑ - 2 π α k)} - \frac{1}{π} \int_{0}^{\infty} e^{- ζ c o s h u} Φ_{β} (u, ϑ) d u .

Multiplying by the remaining factors in (eq:modes), with $ζ = r r^{'} / (2 τ)$ , proves the heat identity.

Step 3: the resolvent. Since $B \geq 0$ ,

(B + s^{2})^{- 1} = \int_{0}^{\infty} e^{- τ s^{2}} e^{- τ B} d τ .

The flat heat integral is $e^{- s^{2} + m^{2} R} / (4 π R)$ . For generic pairs the diffraction weight is bounded at $u = 0$ , and the exponential in $ρ (u)$ controls infinity. After compact spatial smearing, the joint bound in Lemma lem:Phi justifies the same interchanges across image boundaries. This proves the resolvent formula.

Step 4: the global wave distribution. Define the right-hand side of (eq:wave-master) first by joint testing in time and the spatial pair. The geometric terms are the usual flat wave distributions on the indicated angular sectors, with their locally integrable sector indicators.

For the diffracted term, compact time support bounds $u$ : $F_{m}^{+} (t, ρ (u))$ vanishes unless $ρ (u)$ is no larger than the largest positive time in the test support. On compact regular spatial sets, $ρ (u) \geq r + r^{'} > 0$ . After time testing,

⟨ F_{m}^{+} (t, ρ), a (t) ⟩

is a bounded smooth function of $ρ > 0$ on the relevant compact range. Lemma lem:Phi therefore makes the joint $u$ -angular integration locally integrable, including at $u = 0$ and an image boundary. This gives an unambiguous global distribution.

Its Laplace transform is the resolvent formula from Step 3, because

\int_{0}^{\infty} e^{- s t} F_{m}^{+} (t, R) d t = \frac{e ^{- s^{2} + m^{2} R}}{4 π R} .

On the spectral side,

\int_{0}^{\infty} e^{- s t} W_{B} (t) d t = (B + s^{2})^{- 1} .

Both sides are causal distributions of at most polynomial growth after compact spatial smearing. Injectivity of the Laplace transform therefore proves (eq:wave-master) globally.

For $m = 0$ , integrating the delta in $u$ uses

ρ^{'} (u) = \frac{r r ^{'} sinh u}{ρ ( u )} .

This gives (eq:massless). At a generic pair, $Φ_{β}$ is bounded at zero and the resulting edge is of order $(t - ρ_{0})^{- 1/2}$ . Finally, the advanced identity follows from Definition def:Green and oddness of $W_{B}$ .

Displayed formulas in this passage

I-B-G@2#eq:modes · line 1107
$H_{β} (τ; x, x^{'}) = \frac{e ^{- Δ z^{2} / (4 τ)}}{4 π τ} \frac{e ^{- (r^{2} + r^{'2}) / (4 τ)}}{4 π α τ} \times n \in Z \sum e^{i (n + β) (φ - φ^{'})} I_{ν_{n}} (\frac{r r ^{'}}{2 τ}), τ > 0.$
I-B-G@2#eq:flat-massive · line 1130
$F_{m}^{+} (t, R) = \frac{δ ( t - R )}{4 π R} - θ (t - R) \frac{m J _{1} ( m t ^{2} - R ^{2} )}{4 π t ^{2} - R ^{2}},$
I-B-G@2#eq:wave-master · line 1141
$G_{+} (t; x, x^{'}) = S_{β} [F_{m}^{+} (t, \cdot)] .$
I-B-G@2#eq:massless · line 1146
$G_{+} (t; x, x^{'}) = k \in K (ϑ) \sum e^{2 π i β k} \frac{δ ( t - R _{k} )}{4 π R _{k}} - θ (t - ρ_{0}) \frac{Φ _{β} ( u _{0} , ϑ )}{4 π ^{2} α r r ^{'} sinh u _{0}},$
I-B-G@2#eq:Sommerfeld · line 1235
$S (ζ, ϑ) = α k \in K (ϑ) \sum e^{2 π i β k} e^{ζ c o s (ϑ - 2 π α k)} - \frac{1}{π} \int_{0}^{\infty} e^{- ζ c o s h u} Φ_{β} (u, ϑ) d u .$

I-B-G@2#eq:flat-massive · line 1119

\mathcal S_\beta\left[
      (4\pi\tau)^{-3/2}e^{-m^2\tau-R^2/(4\tau)}
                  \right].
\item For \(s>0\), the kernel of \((B+s^2)^{-1}\) is
\[
 \mathcal S_\beta\left[
   \frac{e^{-\sqrt{s^2+m^2}\,R}}{4\pi R}
                 \right].
\item Let \(F_m^+(t,R)\) be the flat retarded kernel of
\(\partial_t^2-\Delta_{\mathbb R^3}+m^2\). For \(R>0\),
\begin{equation}
 F_m^+(t,R)
 =\frac{\delta(t-R)}{4\pi R}
 -\theta(t-R)\,
   \frac{mJ_1(m\sqrt{t^2-R^2})}
        {4\pi\sqrt{t^2-R^2}},
 \label{eq:flat-massive}
\end{equation}
where the second term is zero at \(m=0\), and its
quotient has its continuous value at the front.
The intrinsic retarded kernel is
\begin{equation}
 G_+(t;x,x')=\mathcal S_\beta[F_m^+(t,\cdot)].
 \label{eq:wave-master}
\end{equation}
For \(m=0\), at a generic pair,
\begin{equation}
 G_+(t;x,x')
 =\sum_{k\in\mathcal K(\vartheta)}
     e^{2\pi\mathrm i\beta k}\frac{\delta(t-R_k)}{4\pi R_k}
 -\theta(t-\rho_0)
   \frac{\Phi_\beta(u_0,\vartheta)}
        {4\pi^2\alpha rr'\sinh u_0},
 \label{eq:massless}
\end{equation}
with
\[
 \cosh u_0
 =\frac{t^2-r^2-r'^2-\Delta z^2}{2rr'}.

I-B-G@2#eq:wave-master · line 1119

\mathcal S_\beta\left[
      (4\pi\tau)^{-3/2}e^{-m^2\tau-R^2/(4\tau)}
                  \right].
\item For \(s>0\), the kernel of \((B+s^2)^{-1}\) is
\[
 \mathcal S_\beta\left[
   \frac{e^{-\sqrt{s^2+m^2}\,R}}{4\pi R}
                 \right].
\item Let \(F_m^+(t,R)\) be the flat retarded kernel of
\(\partial_t^2-\Delta_{\mathbb R^3}+m^2\). For \(R>0\),
\begin{equation}
 F_m^+(t,R)
 =\frac{\delta(t-R)}{4\pi R}
 -\theta(t-R)\,
   \frac{mJ_1(m\sqrt{t^2-R^2})}
        {4\pi\sqrt{t^2-R^2}},
 \label{eq:flat-massive}
\end{equation}
where the second term is zero at \(m=0\), and its
quotient has its continuous value at the front.
The intrinsic retarded kernel is
\begin{equation}
 G_+(t;x,x')=\mathcal S_\beta[F_m^+(t,\cdot)].
 \label{eq:wave-master}
\end{equation}
For \(m=0\), at a generic pair,
\begin{equation}
 G_+(t;x,x')
 =\sum_{k\in\mathcal K(\vartheta)}
     e^{2\pi\mathrm i\beta k}\frac{\delta(t-R_k)}{4\pi R_k}
 -\theta(t-\rho_0)
   \frac{\Phi_\beta(u_0,\vartheta)}
        {4\pi^2\alpha rr'\sinh u_0},
 \label{eq:massless}
\end{equation}
with
\[
 \cosh u_0
 =\frac{t^2-r^2-r'^2-\Delta z^2}{2rr'}.

I-B-G@2#eq:massless · line 1119

\mathcal S_\beta\left[
      (4\pi\tau)^{-3/2}e^{-m^2\tau-R^2/(4\tau)}
                  \right].
\item For \(s>0\), the kernel of \((B+s^2)^{-1}\) is
\[
 \mathcal S_\beta\left[
   \frac{e^{-\sqrt{s^2+m^2}\,R}}{4\pi R}
                 \right].
\item Let \(F_m^+(t,R)\) be the flat retarded kernel of
\(\partial_t^2-\Delta_{\mathbb R^3}+m^2\). For \(R>0\),
\begin{equation}
 F_m^+(t,R)
 =\frac{\delta(t-R)}{4\pi R}
 -\theta(t-R)\,
   \frac{mJ_1(m\sqrt{t^2-R^2})}
        {4\pi\sqrt{t^2-R^2}},
 \label{eq:flat-massive}
\end{equation}
where the second term is zero at \(m=0\), and its
quotient has its continuous value at the front.
The intrinsic retarded kernel is
\begin{equation}
 G_+(t;x,x')=\mathcal S_\beta[F_m^+(t,\cdot)].
 \label{eq:wave-master}
\end{equation}
For \(m=0\), at a generic pair,
\begin{equation}
 G_+(t;x,x')
 =\sum_{k\in\mathcal K(\vartheta)}
     e^{2\pi\mathrm i\beta k}\frac{\delta(t-R_k)}{4\pi R_k}
 -\theta(t-\rho_0)
   \frac{\Phi_\beta(u_0,\vartheta)}
        {4\pi^2\alpha rr'\sinh u_0},
 \label{eq:massless}
\end{equation}
with
\[
 \cosh u_0
 =\frac{t^2-r^2-r'^2-\Delta z^2}{2rr'}.

unlabeled · line 1166
$G_{-} (t; x, x^{'}) = G_{+} (- t; x, x^{'}) .$
unlabeled · line 1177
$\int_{0}^{\infty} e^{- τ k^{2}} J_{ν} (k r) J_{ν} (k r^{'}) k d k = \frac{1}{2 τ} e^{- (r^{2} + r^{'2}) / (4 τ)} I_{ν} (\frac{r r ^{'}}{2 τ}) .$
unlabeled · line 1191
$S (ζ, ϑ) = n \sum I_{ν_{n}} (ζ) e^{i (n + β) ϑ / α} .$
unlabeled · line 1197
$I_{ν} (ζ) = \frac{1}{π} \int_{0}^{π} e^{ζ c o s v} cos (ν v) d v - \frac{sin ( π ν )}{π} \int_{0}^{\infty} e^{- ζ c o s h u - ν u} d u,$
unlabeled · line 1207
$h_{ζ} (ϑ) = α k \in Z \sum e^{2 π i β k} e^{ζ c o s (ϑ - 2 π α k)} 1_{{∣ ϑ - 2 π α k ∣ < π}} .$
unlabeled · line 1216
$\frac{1}{2 π} \int_{- π}^{π} e^{ζ c o s v} e^{- i (n + β) v / α} d v .$
unlabeled · line 1227
$- \frac{1}{π} \int_{0}^{\infty} e^{- ζ c o s h u} Φ_{β} (u + ϵ, ϑ) d u .$
unlabeled · line 1250
$(B + s^{2})^{- 1} = \int_{0}^{\infty} e^{- τ s^{2}} e^{- τ B} d τ .$
unlabeled · line 1275
$⟨ F_{m}^{+} (t, ρ), a (t) ⟩$
unlabeled · line 1286
$\int_{0}^{\infty} e^{- s t} F_{m}^{+} (t, R) d t = \frac{e ^{- s^{2} + m^{2} R}}{4 π R} .$
unlabeled · line 1291
$\int_{0}^{\infty} e^{- s t} W_{B} (t) d t = (B + s^{2})^{- 1} .$
unlabeled · line 1301
$ρ^{'} (u) = \frac{r r ^{'} sinh u}{ρ ( u )} .$

Theorem: Heat, resolvent, and wave representation

Displayed formulas in this passage

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