$I_{2,2}^{(1,1)}(x,y)=\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z\chi_{0}(z^{2})\frac{e^{iz|x-x_{1}|}}{-2iz}[\underline{e}_{11}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\frac{e^{iz|y-y_{1}|}}{-2iz}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z.$

Using the orthogonality (5.43), we have

$\begin{split}I_{2,2}^{(1,1)}(x,y)&=\frac{1}{4}\int_{{\mathbb{R}}^{3}}\int_{|x|}^{|x-x_{1}|}\int_{|y|}^{|y-y_{1}|}e^{itz^{2}+iz(s_{1}+s_{2})}z\chi_{0}(z^{2})[\underline{e}_{11}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\,\mathrm{d}s_{1}\,\mathrm{d}s_{2}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&\quad+\frac{1}{4i}\int_{{\mathbb{R}}^{3}}\int_{|x|}^{|x-x_{1}|}e^{itz^{2}+izs_{1}}\chi_{0}(z^{2})[\underline{e}_{11}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})e^{iz|y|}\,\mathrm{d}s_{1}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&=:I_{2,2;1}^{(1,1)}+I_{2,2;2}^{(1,1)}.\end{split}$

By Lemma 2.1, we have

$\left|\int_{{\mathbb{R}}}e^{itz^{2}+iz(s_{1}+s_{2})}z\chi_{0}(z^{2})\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle s_{1}\rangle\langle s_{2}\rangle.$

Using this estimate, the bound

$\int_{|x|}^{|x-x_{1}|}\int_{|y|}^{|y-y_{1}|}\langle s_{1}\rangle\langle
s_{2}\rangle\,\mathrm{d}s_{1}\,\mathrm{d}s_{2}\lesssim\langle
x_{1}\rangle^{2}\langle y_{2}\rangle^{2}\langle x\rangle\langle y\rangle,$

the absolute boundedness of $S_{1}TP$, and the exponential decay of
$v_{1},v_{2}$, we deduce that

$\begin{split}\left|I_{2,2;1}^{(1,1)}(x,y)\right|\lesssim|t|^{-\frac{3}{2}}\langle
x\rangle\langle y\rangle\int_{{\mathbb{R}}^{2}}|\langle
x_{1}\rangle^{2}\langle
y_{2}\rangle^{2}[\underline{e}_{11}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})|\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\lesssim|t|^{-\frac{3}{2}}\langle
x\rangle\langle y\rangle.\end{split}$ (5.60)

By Lemma 4.5 and direct computation,

$\int_{{\mathbb{R}}}S_{1}TP(x_{1},y_{1})v_{2}(y_{1})\underline{e}_{11}\,\mathrm{d}y_{1}=\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}\Phi(x_{1})[c_{0}\underline{e}_{1}]^{*}.$
(5.61)

Hence, integrating in $y_{1}$, we have

$\begin{split}I_{2,2;2}^{(1,1)}(x,y)&=\frac{\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}}{4i}\left(\int_{{\mathbb{R}}}\int_{|x|}^{|x-x_{1}|}\int_{{\mathbb{R}}}e^{itz^{2}+iz(s_{1}+|y|)}\chi_{0}(z^{2})\underline{e}_{11}v_{1}(x_{1})\Phi(x_{1})\,\mathrm{d}z\,\mathrm{d}s_{1}\,\mathrm{d}x_{1}\right)[c_{0}\underline{e}_{1}]^{*}\\\
&=\frac{\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}}{4i}\left(\int_{{\mathbb{R}}}\int_{|x|}^{|x-x_{1}|}G_{t}(s_{1}+|y|)\,\mathrm{d}s_{1}\underline{e}_{11}v_{1}(x_{1})\Phi(x_{1})\,\mathrm{d}x_{1}\right)[c_{0}\underline{e}_{1}]^{*},\end{split}$

where $G_{t}$ is the function defined in Lemma 2.2. By Lemma 2.2 (c.f.
(5.44)–(5.48) for similar computations), we have

$\left|I_{2,2;2}^{(1,1)}(x,y)-\frac{i\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}}{2}[G_{0}(\underline{e}_{11}v_{1}\Phi)(x)][e^{i\frac{y^{2}}{4t}}{c_{0}}\underline{e}_{1}]^{*}\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle^{2}\langle y\rangle^{2},$

where $G_{0}$ is the operator defined in (5.49). This completes the analysis
of the term $I_{2,2}^{(1,1)}$. Next, we treat the term

$\begin{split}I_{2,2}^{(2,1)}(x,y)=\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{-2\sqrt{z^{2}+2\mu}}[\underline{e}_{22}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\frac{e^{iz|y-y_{1}|}}{-2iz}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z.\end{split}$
(5.62)

By inserting $e^{iz|y|}$, we write

$\begin{split}I_{2,2}^{(2,1)}(x,y)&=-\frac{1}{2}\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{-2\sqrt{z^{2}+2\mu}}[\underline{e}_{22}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\int_{|y|}^{|y-y_{1}|}e^{izs_{2}}\,\mathrm{d}s_{2}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&\quad+\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{-2\sqrt{z^{2}+2\mu}}[\underline{e}_{22}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\frac{e^{iz|y|}}{-2iz}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&=:I_{2,2;1}^{(2,1)}(x,y)+I_{2,2;2}^{(2,1)}(x,y),\end{split}$

where $I_{2,2;2}^{(2,1)}$ is the leading term. By Lemma 2.1 and Lemma 5.1,

$\left|\int_{{\mathbb{R}}}e^{itz^{2}+izs_{2}}z\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{-2\sqrt{z^{2}+2\mu}}\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle s_{2}\rangle.$ (5.63)

Hence, using the absolute boundedness of $S_{1}TP$ and the bound (5.45), we
have

$\begin{split}\left|I_{2,2;1}^{(2,1)}(x,y)\right|\lesssim|t|^{-\frac{3}{2}}\int_{{\mathbb{R}}^{2}}\langle
y_{1}\rangle^{2}\langle
y\rangle[\underline{e}_{22}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\lesssim|t|^{-\frac{3}{2}}\langle
y\rangle.\end{split}$

On the other hand, we treat $I_{2,2;1}^{(2,1)}$ similarly as in (5.50) -
(5.53) and find that

$\begin{split}\left|I_{2,2;2}^{(2,1)}(x,y)-\frac{i}{2}\int_{{\mathbb{R}}^{3}}e^{itz^{2}+iz|y|}\chi_{0}(z^{2})G_{2}(x,x_{1})[\underline{e}_{22}v_{1}S_{1}TPv_{2}\underline{e}_{11}](x_{1},y_{1})\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle y\rangle,\end{split}$

where $G_{2}$ is defined in (5.51). Hence, by Lemma 2.2 and (5.61), we
conclude that

$\left|I_{2,2}^{(2,1)}(x,y)-\frac{i\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}}{2}\frac{\sqrt{\pi}}{{\sqrt{-it}}}[G_{2}(\underline{e}_{22}v_{1}\Phi)(x)][e^{i\frac{y^{2}}{4t}}{c_{0}}\underline{e}_{1}]^{*}\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle y\rangle.$ (5.64)

Finally, we show that the terms $I_{2,2}^{(1,2)}$ and $I_{2,2}^{(2,2)}$
satisfy the better decay rates of $\mathcal{O}(|t|^{-\frac{3}{2}}\langle
x\rangle\langle y\rangle)$. By orthogonality (c.f. (5.43)),

$\begin{split}&I_{2,2}^{(1,2)}(x,y)\\\
&=\frac{1}{-2}\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z\chi_{0}(z^{2})\int_{|x|}^{|x-x_{1}|}e^{izs_{1}}\,\mathrm{d}s_{1}[\underline{e}_{11}v_{1}S_{1}TPv_{2}\underline{e}_{22}](x_{1},y_{1})\frac{e^{-\sqrt{z^{2}+2\mu}|y-y_{1}|}}{-2\sqrt{z^{2}+2\mu}}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z.\end{split}$

By Lemma 2.1 and Lemma 5.1, we note that the $z$-integral satisfy the bound

$\left|\int_{{\mathbb{R}}}e^{itz^{2}+izs_{1}}z\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|y-y_{1}|}}{-2\sqrt{z^{2}+2\mu}}\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle s_{1}\rangle.$

Hence, by the absolute boundedness of $S_{1}TP$ and decay of $v_{1}$, $v_{2}$,
we conclude that

$\left|I_{2,2}^{(1,2)}(x,y)\right|\leq C|t|^{-\frac{3}{2}}\langle x\rangle.$

The analysis of $I_{2,2}^{(2,2)}$ is analogous to the preceeding one, yielding
the bound

$\left|I_{2,2}^{(2,2)}(x,y)\right|\leq C|t|^{-\frac{3}{2}}\langle y\rangle.$

Thus, using $\mathcal{G}_{0}=G_{0}\underline{e}_{11}+G_{2}\underline{e}_{22}$,
and $\mathcal{G}_{0}(v_{1}\Phi)=c_{0}\underline{e}_{1}-\Psi$ from Lemma 4.5,
we conclude (5.56) and (5.58). For the estimate (5.57) involving $I_{2,3}$,
one should instead use the identity

$\int_{{\mathbb{R}}}\underline{e}_{11}v_{1}(x_{1})PTS_{1}(x_{1},y_{1})\,\mathrm{d}x_{1}=-\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}c_{0}\underline{e}_{1}\Phi(y_{1})^{*},$
(5.65)

and we leave the remaining details to the reader. ∎

Next, we remark that the analysis for $I_{2,4}$ involving the operator $P$
leads to a similar estimate as the free evolution in Proposition 2.3.

######  Proposition 5.6.

For all $|t|\geq 1$, we have

$\left|I_{2,4}(x,y)-F_{t}^{4}(x,y)\right|\leq C|t|^{-\frac{3}{2}}\langle
x\rangle^{2}\langle y\rangle^{2},$ (5.66)

where

$F_{t}^{4}(x,y):=\frac{\|V_{1}\|_{L^{1}({\mathbb{R}})}}{4}\frac{\sqrt{\pi}}{{\sqrt{-it}}}e^{-i\frac{x^{2}}{4t}}\underline{e}_{1}e^{-i\frac{y^{2}}{4t}}\underline{e}_{1}^{\top}.$
(5.67)

###### Proof.

As before, we write

$I_{2,4}=I_{2,4}^{(1,1)}+I_{2,4}^{(1,2)}+I_{2,4}^{(2,1)}+I_{2,4}^{(2,2)},$

with

$I_{2,4}^{(i,j)}:=\int_{{\mathbb{R}}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})[\mathcal{R}_{i}(z)v_{1}Pv_{2}\mathcal{R}_{j}(z)]\,\mathrm{d}z,\quad
i,j\in\\{1,2\\},$

where $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ were defined in (5.11). We first
treat the leading term

$I_{2,4}^{(1,1)}(x,y)=\int_{{\mathbb{R}}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\frac{e^{iz|x-x_{1}|}}{2iz}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{11}](x_{1},y_{1})\frac{e^{iz|y-y_{1}|}}{2iz}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z.$
(5.68)

By adding and subtracting $e^{iz|x|}$ and $e^{iz|y|}$ twice, we further
consider

$\begin{split}I_{2,4}^{(1,1)}(x,y)&=\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\frac{e^{iz|x|}}{2iz}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{11}](x_{1},y_{1})\frac{e^{iz|y|}}{2iz}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&\quad+\frac{1}{2}\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\frac{e^{iz|x|}}{2iz}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{11}](x_{1},y_{1})\int_{|y|}^{|y-y_{1}|}e^{izs_{2}}\,\mathrm{d}s_{2}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&\quad+\frac{1}{2}\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\int_{|x|}^{|x-x_{1}|}e^{izs_{1}}ds_{1}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{11}](x_{1},y_{1})\frac{e^{iz|y|}}{2iz}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&\quad+\frac{1}{4}\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\int_{|x|}^{|x-x_{1}|}e^{izs_{1}}ds_{1}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{11}](x_{1},y_{1})\int_{|y|}^{|y-y_{1}|}e^{izs_{2}}\,\mathrm{d}s_{2}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z\\\
&=:I_{2,4;1}^{(1,1)}(x,y)+I_{2,4;2}^{(1,1)}(x,y)+I_{2,4;3}^{(1,1)}(x,y)+I_{2,4;4}^{(1,1)}(x,y).\end{split}$

By direct computation,

$\int_{{\mathbb{R}}^{2}}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{11}](x_{1},y_{1})\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}=-\|V_{1}\|_{L^{1}({\mathbb{R}})}\underline{e}_{1}\underline{e}_{1}^{\top}.$

Hence, by Lemma 2.2,

$\left|I_{2,4;1}^{(1,1)}(x,y)-\frac{\|V_{1}\|_{L^{1}({\mathbb{R}})}}{4}\frac{\sqrt{\pi}}{{\sqrt{-it}}}e^{-i\frac{x^{2}}{4t}}\underline{e}_{1}e^{-i\frac{y^{2}}{4t}}\underline{e}_{1}^{\top}\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle y\rangle.$

For the terms $I_{2,4;2}^{(1,1)}$, $I_{2,4;3}^{(1,1)}$, the additional factor
of $z$ allows to invoke Lemma 2.1,

$\begin{split}\left|\int_{{\mathbb{R}}}e^{itz^{2}+iz(|x|+s_{2})}z\chi_{0}(z^{2})\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle s_{2}\rangle,\\\
\left|\int_{{\mathbb{R}}}e^{itz^{2}+iz(s_{1}+|y|)}z\chi_{0}(z^{2})\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle y\rangle\langle s_{1}\rangle.\end{split}$

Thus, we infer from the exponential decay of $v_{1}$ and $v_{2}$ that

$\left|I_{2,4;2}^{(1,1)}(x,y)\right|+\left|I_{2,4;3}^{(1,1)}(x,y)\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle y\rangle.$ (5.69)

For the term $I_{2,4;4}^{(1,1)}$, we can use non-stationary phase to conclude
the same bound. Hence, we have

$\left|I_{2,4}^{(1,1)}(x,y)-\frac{\|V_{1}\|_{L^{1}({\mathbb{R}})}}{4}\frac{\sqrt{\pi}}{{\sqrt{-it}}}e^{-i\frac{x^{2}}{4t}}\underline{e}_{1}e^{-i\frac{y^{2}}{4t}}\underline{e}_{1}^{\top}\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle y\rangle.$

Thus, it remains to prove that the other terms $I_{2,4}^{(1,2)}$,
$I_{2,4}^{(2,1)}$, $I_{2,4}^{(2,2)}$ have the better
$\mathcal{O}(|t|^{-\frac{3}{2}}\langle x\rangle\langle y\rangle)$ weighted
decay estimate to finish the proposition. We first treat the term

$\begin{split}I_{2,4}^{(1,2)}(x,y)&=\frac{1}{2i}\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z\chi_{0}(z^{2})e^{iz|x-x_{1}|}[\underline{e}_{11}v_{1}Pv_{2}\underline{e}_{22}](x_{1},y_{1})\frac{e^{-\sqrt{z^{2}+2\mu}|y-y_{1}|}}{-2\sqrt{z^{2}+2\mu}}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z.\\\
\end{split}$ (5.70)

By Lemma 2.1 and Lemma 5.1,

$\left|\int_{{\mathbb{R}}}e^{itz^{2}+iz(|x-x_{1}|)}z\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|y-y_{1}|}}{-2\sqrt{z^{2}+2\mu}}\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\langle x\rangle\langle x_{1}\rangle.$

Hence, using the decay assumptions on $v_{1}$ and $v_{2}$, we conclude that

$\left|I_{2,4}^{(1,2)}(x,y)\right|\leq C|t|^{-\frac{3}{2}}\langle
x\rangle\langle y\rangle.$

The same bound holds for the term $I_{2,4}^{(2,1)}$ and we will skip the
details. Finally, we are left with

$I_{2,4}^{(2,2)}(x,y)=\int_{{\mathbb{R}}^{3}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{-2\sqrt{z^{2}+2\mu}}[\underline{e}_{22}v_{1}Pv_{2}\underline{e}_{22}](x_{1},y_{1})\frac{e^{-\sqrt{z^{2}+2\mu}|y-y_{1}|}}{-2\sqrt{z^{2}+2\mu}}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\,\mathrm{d}z.$

By direct computation using (3.2),

$[\underline{e}_{22}v_{1}Pv_{2}\underline{e}_{22}](x_{1},y_{1})=\frac{1}{\|V_{1}\|_{L^{1}({\mathbb{R}})}}[V_{2}\underline{e}_{2}](x_{1})[V_{2}\underline{e}_{2}]^{\top}(y_{1}),$

and by Lemma 2.1 and Lemma 5.1, we have the uniform estimate

$\left|\int_{{\mathbb{R}}}e^{itz^{2}}z^{2}\chi_{0}(z^{2})\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{-2\sqrt{z^{2}+2\mu}}\frac{e^{-\sqrt{z^{2}+2\mu}|y-y_{1}|}}{-2\sqrt{z^{2}+2\mu}}dz\right|\leq
C_{\mu}|t|^{-\frac{3}{2}}.$

Hence, by exchanging the order of integration, we conclude that

$\left|I_{2,4}^{(2,2)}(x,y)\right|\leq C|t|^{-\frac{3}{2}}.$ (5.71)

Thus, we conclude (5.66) by summing over the four terms. ∎

Finally, we are ready to complete the proof of the local decay estimate (5.8).
We sum the leading contributions of the spectral representation of
$e^{it\mathcal{H}}\chi_{0}(\mathcal{H}-\mu I)P_{\mathrm{s}}^{+}$ in (5.24) by
invoking Proposition 2.3, Proposition 5.4, Proposition 5.5, and Proposition
5.6 to obtain

$\begin{split}&F_{t}^{0}-\frac{e^{it\mu}}{\pi
i}\left(\frac{i}{2\eta}F_{t}^{1}+\frac{1}{\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}}F_{t}^{2}+\frac{1}{\eta\|V_{1}\|_{L^{1}({\mathbb{R}})}}F_{t}^{3}+\left(\frac{2i}{\|V_{1}\|_{L^{1}({\mathbb{R}})}}+\frac{2|c_{0}|^{2}}{i\|V_{1}\|_{L^{1}({\mathbb{R}})}}\right)F_{t}^{4}\right)\\\
&=\frac{e^{it\mu}}{\sqrt{-4\pi
it}}\left(-[c_{0}\underline{e}_{1}-\Psi(x)][\sigma_{3}\Psi(y)-{c_{0}}\underline{e}_{1}]^{*}-[c_{0}\underline{e}_{1}-\Psi(x)][e^{i\frac{y^{2}}{4t}}{c_{0}}\underline{e}_{1}]^{*}\right.\\\
&\left.\hskip
142.26378pt+[e^{-i\frac{x^{2}}{4t}}c_{0}\underline{e}_{1}][\sigma_{3}\Psi(y)-{c_{0}}\underline{e}_{1}]^{*}+|c_{0}|^{2}e^{-i\frac{x^{2}}{4t}}e^{-i\frac{y^{2}}{4t}}\underline{e}_{1}\underline{e}_{1}^{\top}\right)\\\
&=\frac{e^{it\mu}}{\sqrt{-4\pi
it}}\left(\Psi(x)[\sigma_{3}\Psi(y)]^{*}+(e^{-i\frac{x^{2}}{4t}}-1)c_{0}[\sigma_{3}\Psi(y)]^{*}+(e^{-i\frac{y^{2}}{4t}}-1)\Psi(x)[{c_{0}}\underline{e}_{1}]^{*}\right.\\\
&\left.\hskip
142.26378pt+(1-e^{-i\frac{x^{2}}{4t}}-e^{-i\frac{y^{2}}{4t}}+e^{-i\frac{x^{2}}{4t}}e^{-i\frac{y^{2}}{4t}})|c_{0}|^{2}\underline{e}_{1}\underline{e}_{1}^{\top}\right),\end{split}$

where we use the cancellation $F_{t}^{0}-\frac{e^{it\mu}}{\pi
i}\frac{2i}{\|V_{1}\|_{L^{1}({\mathbb{R}})}}F_{t}^{4}=0$ in the first
equality. We note that the first term gives us the finite rank operator

$F_{t}^{+}(x,y)=\frac{e^{it\mu}}{\sqrt{-4\pi
it}}\Psi(x)[\sigma_{3}\Psi(y)]^{*},$ (5.72)

and we show that the last three terms satisfy the better decay rate. Using,

$|1-e^{-i\frac{x^{2}}{4t}}|\leq\frac{x^{2}}{4|t|},$ (5.73)

and the fact that $\Psi\in L^{\infty}({\mathbb{R}})\times
L^{\infty}({\mathbb{R}})$, we have

$\left|\frac{e^{it\mu}e^{i\frac{\pi}{4}}}{2\sqrt{\pi}\sqrt{t}}(e^{-i\frac{x^{2}}{4t}}-1)c_{0}\underline{e}_{1}[\sigma_{3}\Psi(y)]^{*}\right|\lesssim|t|^{-\frac{3}{2}}\langle
x\rangle^{2},$

and similarly

$\left|\frac{e^{it\mu}e^{i\frac{\pi}{4}}}{2\sqrt{\pi}\sqrt{t}}(e^{-i\frac{y^{2}}{4t}}-1)\overline{c_{0}}\Psi(x)\underline{e}_{1}^{\top}\right|\lesssim|t|^{-\frac{3}{2}}\langle
y\rangle^{2}.$

For the last term, we have

$\left|1-e^{-i\frac{x^{2}}{4t}}-e^{-i\frac{y^{2}}{4t}}+e^{-i\frac{x^{2}}{4t}}e^{-i\frac{y^{2}}{4t}}\right|=\left|1-e^{-i\frac{x^{2}}{4t}}\right|\left|1-e^{-i\frac{y^{2}}{4t}}\right|\lesssim|t|^{-2}\langle
x\rangle^{2}\langle y\rangle^{2}.$ (5.74)

Thus, the leading contribution to $e^{it\mathcal{H}}\chi_{0}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}$ is $F_{t}^{+}$. ∎

##  6\. Intermediate and high energy estimates

In order to complete the proof of Theorem 1.4, we also need to prove the
dispersive estimates when the spectral variable is bounded away from the
thresholds $\pm\mu$. As usual, we focus on the positive semi-axis
$[\mu,\infty)$ of the essential spectrum and prove the dispersive estimates
for energies $\lambda>\mu$. The negative semi-axis $(-\infty,-\mu]$ can be
treated by symmetry of $\mathcal{H}$. We recall from Section 2 that the kernel
of the limiting resolvent operator for $\mathcal{H}_{0}$ has the formula

$\mathcal{R}_{0}^{\pm}(z)(x,y):=(\mathcal{H}_{0}-(z^{2}+\mu\pm
i0))^{-1}=\begin{bmatrix}\pm\frac{ie^{\pm iz|x-y|}}{2z}&0\\\
0&-\frac{e^{-\sqrt{z^{2}+2\mu}|x-y|}}{2\sqrt{z^{2}+2\mu}}\end{bmatrix},\quad\forall\
0<z<\infty.$ (6.1)

From this, we have the following bound

$\|\mathcal{R}_{0}^{\pm}(z)\|_{L^{1}\times L^{1}\to L^{\infty}\times
L^{\infty}}\leq C|z|^{-1}.$

Hence, for sufficiently large $z$, the perturbed resolvent
$\mathcal{R}^{\pm}(z)$ can be expanded into the infinite Born series

$\mathcal{R}^{\pm}(z)=\sum_{n=0}^{\infty}\mathcal{R}_{0}^{\pm}(z)(-\mathcal{V}\mathcal{R}_{0}^{\pm}(z))^{n}.$
(6.2)

More precisely, since the operator norm $L^{1}\times L^{1}\to L^{\infty}\times
L^{\infty}$ in the $n$-th summand above is bounded by
$C|z|^{-1}(C\|\mathcal{V}\|_{1}|z|^{-1})^{n}$, the Born series converges in
the operator norm whenever $|z|>z_{1}:=2C\|\mathcal{V}\|_{L^{1}\times L^{1}}$.
We define the high-energy cut-off by

$\chi_{\mathrm{h}}(z):=1-\chi(z),$ (6.3)

where $\chi(z)$ is a standard smooth even cut-off supported on
$[-z_{1},z_{1}]$ satisfying $\chi(z)=1$ for $|z|\leq\frac{z_{1}}{2}$ and
$\chi(z)=0$ for $|z|\geq z_{1}$. We insert the cut-off and the Born series
expansion into the spectral representation
$e^{it\mathcal{H}}\chi_{\mathrm{h}}(\mathcal{H}-\mu I)P_{\mathrm{s}}^{+}$ and
look to bound the following

$\begin{split}|\langle e^{it\mathcal{H}}\chi_{\mathrm{h}}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}\vec{u},\vec{v}\rangle|&=\left|\int_{0}^{\infty}e^{itz^{2}}z\chi_{\mathrm{h}}(z^{2})\langle[\mathcal{R}^{+}(z)-\mathcal{R}^{-}(z)]\vec{u},\vec{v}\rangle\,\mathrm{d}z\right|\\\
&\leq
C\sum_{\pm}\sum_{n=0}^{\infty}\left|\int_{0}^{\infty}e^{itz^{2}}z\chi_{\mathrm{h}}(z^{2})\langle\mathcal{R}_{0}^{\pm}(z)(\mathcal{V}\mathcal{R}_{0}^{\pm}(z))^{n}\vec{u},\vec{v}\rangle\,\mathrm{d}z\right|,\end{split}$
(6.4)

where
$\vec{u},\vec{v}\in\mathcal{S}({\mathbb{R}})\times\mathcal{S}({\mathbb{R}})$.
From [KS06], we have the following dispersive estimates:

######  Proposition 6.1.

Under the same hypothesis as Theorem 1.4, we have

$\left\|e^{it\mathcal{H}}\chi_{\mathrm{h}}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}\vec{u}\,\right\|_{L^{\infty}({\mathbb{R}})\times
L^{\infty}({\mathbb{R}})}\lesssim|t|^{-\frac{1}{2}}\left\|\vec{u}\,\right\|_{L^{1}({\mathbb{R}})\times
L^{1}({\mathbb{R}})},$ (6.5)

and

$\left\|\langle
x\rangle^{-1}e^{it\mathcal{H}}\chi_{\mathrm{h}}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}\vec{u}\,\right\|_{L^{\infty}({\mathbb{R}})\times
L^{\infty}({\mathbb{R}})}\lesssim|t|^{-\frac{3}{2}}\left\|\langle
x\rangle\vec{u}\,\right\|_{L^{1}({\mathbb{R}})\times L^{1}({\mathbb{R}})},$
(6.6)

for any $|t|\geq 1$.

###### Proof.

For (6.5), see the proof of [KS06, Proposition 7.1], and for (6.6), see the
proof of [KS06, Proposition 8.1]. Note that the high-energy dispersive
estimate holds irrespective of the regularity of the thresholds $\pm\mu$. ∎

Let $z_{0}>0$ be the constant from Proposition 4.8. It may happen that $z_{1}$
is strictly larger than $z_{0}$. In this case, we need to derive estimates
analogous to the above proposition in the remaining intermediate energy regime
$[-z_{1},-z_{0}]\cup[z_{0},z_{1}]$. To this end, we set $\chi_{\mathrm{m}}(z)$
to be the intermediate energy cut-off given by

$\chi_{\mathrm{m}}(z):=1-\chi_{0}(z)-\chi_{\mathrm{h}}(z),$ (6.7)

where $\chi_{0}(z)$ was the cut-off defined in the previous section in
Proposition 5.2.

######  Proposition 6.2.

For any $|t|\geq 1$, we have

$\left\|e^{it\mathcal{H}}\chi_{\mathrm{m}}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}\vec{u}\,\right\|_{L_{x}^{\infty}({\mathbb{R}})\times
L_{x}^{\infty}({\mathbb{R}})}\lesssim|t|^{-\frac{1}{2}}\left\|\vec{u}\,\right\|_{L_{x}^{1}({\mathbb{R}})\times
L_{x}^{1}({\mathbb{R}})},$ (6.8)

and

$\left\|\langle
x\rangle^{-1}e^{it\mathcal{H}}\chi_{\mathrm{m}}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}\vec{u}\,\right\|_{L_{x}^{\infty}({\mathbb{R}})\times
L_{x}^{\infty}({\mathbb{R}})}\lesssim|t|^{-\frac{3}{2}}\left\|\langle
x\rangle\vec{u}\,\right\|_{L_{x}^{1}({\mathbb{R}})\times
L_{x}^{1}({\mathbb{R}})}.$ (6.9)

Before proving the above proposition, we need the following lemmas for
pointwise bounds and operator norm bounds on the resolvent operators and its
derivatives. The first lemma follows immediately from the expression (6.1) and
the triangle inequality $||x-x_{1}|-|x||\leq|x_{1}|$.

######  Lemma 6.3.

Let $\gamma_{0}>0$. For every $z>\gamma_{0}$, and $k\in\\{0,1,2\\}$, we have

$\left|\partial_{z}^{k}\mathcal{R}_{0}^{\pm}(z)(x,y)\right|\leq
C\gamma_{0}^{-1-k}\langle x-y\rangle^{k},$ (6.10)

and hence

$\left\|\partial_{z}^{k}\mathcal{R}_{0}^{\pm}(z)(x,\cdot)\right\|_{X_{-(\frac{1}{2}+k)-}}\leq
C\gamma_{0}^{-1-k}\langle x\rangle^{k}.$ (6.11)

Moreover, define

$\mathcal{G}_{\pm}(z)(x,x_{1})=\begin{bmatrix}e^{\mp iz|x|}&0\\\
0&1\end{bmatrix}\mathcal{R}_{0}^{\pm}(z)(x,x_{1})=\begin{bmatrix}\pm\frac{ie^{\pm
iz(|x-x_{1}|-|x|)}}{2z}&0\\\
0&-\frac{e^{-\sqrt{z^{2}+2\mu}|x-x_{1}|}}{2\sqrt{z^{2}+2\mu}}\end{bmatrix}.$
(6.12)

Then, for any $k\geq 0$,

$\sup_{x\in{\mathbb{R}}}\left|\partial_{z}^{k}\mathcal{G}^{\pm}(z)(x,x_{1})\right|\leq
C\gamma_{0}^{-1-k}|x_{1}|.$ (6.13)

With these bounds, we are able to give operator norm bounds on the perturbed
resolvent via the resolvent identity.

######  Lemma 6.4.

Let $\gamma_{0}>0$. We have

$\sup_{|z|>\gamma_{0}}\left\|\partial_{z}\mathcal{R}^{\pm}(z)\right\|_{X_{\frac{3}{2}+}\to
X_{-\frac{3}{2}-}}\lesssim 1,$ (6.14)
$\sup_{|z|>\gamma_{0}}\left\|\partial_{z}^{2}\mathcal{R}^{\pm}(z)\right\|_{X_{\frac{5}{2}+}\to
X_{-\frac{5}{2}-}}\lesssim 1.$ (6.15)

###### Proof.

By Lemma 3.1, for any $|z|>\gamma_{0}$, we have

$\mathcal{R}^{\pm}(z)=(I+\mathcal{R}_{0}^{\pm}(z)\mathcal{V})^{-1}\mathcal{R}_{0}^{\pm}(z)=:S^{\pm}(z)^{-1}\mathcal{R}_{0}^{\pm}(z),$
(6.16)

as a bounded operator from $X_{\frac{1}{2}+}$ to $X_{-\frac{1}{2}-}$. Note
that $S^{\pm}(z)$ is boundedly invertible on $X_{-\sigma}$ for any $\sigma>0$.
By differentiation, we have

$\partial_{z}\mathcal{R}^{\pm}(z)=-S^{\pm}(z)^{-1}\partial_{z}\mathcal{R}_{0}^{\pm}(z)\mathcal{V}S^{\pm}(z)^{-1}\mathcal{R}_{0}^{\pm}(z)+S^{\pm}(z)^{-1}\partial_{z}\mathcal{R}_{0}^{\pm}(z).$
(6.17)

Moreover, as a multiplication operator, $\mathcal{V}:X_{-\sigma}\to
X_{\sigma}$ is bounded for any $\sigma>0$ due to the exponential decay of
$\mathcal{V}$. By Lemma 6.3, $\partial_{z}R_{0}^{\pm}(z):X_{\frac{3}{2}+}\to
X_{-\frac{3}{2}-}$ is bounded and since the embedding
$X_{-\frac{1}{2}-}\subset X_{-\frac{3}{2}-}$ is continuous, we infer the bound
(6.14) by taking composition. By a similar argument,

$\|\partial_{z}^{2}\mathcal{R}^{\pm}(z)\|_{X_{\frac{5}{2}+}\to
X_{-\frac{5}{2}-}}\lesssim 1.$ (6.18)

∎

###### Proof of Proposition 6.2.

By iterating the second resolvent identity, we write the perturbed resolvent
as a finite sum

$\mathcal{R}^{\pm}(z)=\mathcal{R}_{0}^{\pm}(z)-\mathcal{R}_{0}^{\pm}(z)\mathcal{V}\mathcal{R}_{0}^{\pm}(z)+\mathcal{R}_{0}^{\pm}(z)\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{R}_{0}^{\pm}(z),$
(6.19)

and we write

$e^{it\mathcal{H}}\chi_{\mathrm{m}}(\mathcal{H}-\mu
I)P_{\mathrm{s}}^{+}(x,y)=\sum_{j=1}^{3}\int_{0}^{\infty}e^{itz^{2}}z\chi_{\mathrm{m}}(z^{2})(-1)^{j+1}(\mathcal{E}_{j}^{+}(z)-\mathcal{E}_{j}^{-}(z))(x,y)dz,$
(6.20)

with

$\mathcal{E}_{1}^{\pm}(z)=\mathcal{R}_{0}^{\pm}(z),\quad\mathcal{E}_{2}^{\pm}(z)=\mathcal{R}_{0}^{\pm}(z)\mathcal{V}\mathcal{R}_{0}^{\pm}(z),\quad\mathcal{E}_{3}^{\pm}(z)=\mathcal{R}_{0}^{\pm}(z)\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{R}_{0}^{\pm}(z).$
(6.21)

Hence, to prove (6.8) and (6.9), it is sufficient to establish the estimates

$\sup_{\pm}\sup_{j=1,2,3}\left|\int_{0}^{\infty}e^{itz^{2}}z\chi_{\mathrm{m}}(z^{2})\mathcal{E}_{j}^{\pm}(z)(x,y)\,\mathrm{d}z\right|\lesssim\min\\{|t|^{-\frac{1}{2}},|t|^{-\frac{3}{2}}\langle
x\rangle\langle y\rangle\\}.$ (6.22)

The term involving $\mathcal{E}_{1}^{\pm}$ is handled by the earlier
Proposition 2.3, while the second term involving $\mathcal{E}_{2}^{\pm}$ can
be treated analogously as in Proposition 6.1. We refer the reader to [GS04,
Lemma 3] and [Gol07, Proposition 3] for similar computations. For the term
involving $\mathcal{E}_{3}^{\pm}$, we first write

$\mathcal{R}_{0}^{\pm}(z)(s_{1},s_{2})=\begin{bmatrix}e^{\pm iz|s_{1}|}&0\\\
0&1\end{bmatrix}\mathcal{G}_{\pm}(z)(s_{1},s_{2}),$

where the operator $\mathcal{G}_{\pm}(z)$ was defined in (6.12). Then, using
that the kernel $\mathcal{R}_{0}^{\pm}(z)(x,y)$ is symmetric in $x$ and $y$
variables, and using the matrix identity

$\underline{e}_{jj}\begin{bmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{bmatrix}\underline{e}_{kk}=a_{jk}\underline{e}_{j}\underline{e}_{k}^{\top},\quad
j,k\in\\{1,2\\},$ (6.23)

we compute the following kernel identity

$\begin{split}\mathcal{E}_{3}^{\pm}(z)(x,y)&=\int_{{\mathbb{R}}^{2}}\mathcal{R}_{0}^{\pm}(x,x_{1})[\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}](x_{1},y_{1})\mathcal{R}_{0}^{\pm}(y,y_{1})\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\\\
&=\begin{bmatrix}e^{\pm iz|x|}&0\\\
0&1\end{bmatrix}\int_{{\mathbb{R}}^{2}}\mathcal{G}^{\pm}(x,x_{1})[\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}](x_{1},y_{1})\mathcal{G}^{\pm}(y,y_{1})\,\mathrm{d}x_{1}\,\mathrm{d}y_{1}\begin{bmatrix}e^{\pm
iz|y|}&0\\\ 0&1\end{bmatrix}\\\ &=e^{\pm
iz(|x|+|y|)}\langle(\mathcal{G}^{\pm})^{*}(z)(x,\cdot)\underline{e}_{1},\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{G}^{\pm}(z)(y,\cdot)\underline{e}_{1}\rangle\,\underline{e}_{1}\underline{e}_{1}^{\top}\\\
&\quad+e^{\pm
iz|x|}\langle(\mathcal{G}^{\pm})^{*}(z)(x,\cdot)\underline{e}_{2},\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{G}^{\pm}(z)(y,\cdot)\underline{e}_{1}\rangle\,\underline{e}_{1}\underline{e}_{2}^{\top}\\\
&\quad+e^{\pm
iz|y|}\langle(\mathcal{G}^{\pm})^{*}(z)(x,\cdot)\underline{e}_{1},\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{G}^{\pm}(z)(y,\cdot)\underline{e}_{2}\rangle\,\underline{e}_{2}\underline{e}_{1}^{\top}\\\
&\quad+\langle(\mathcal{G}^{\pm})^{*}(z)(x,\cdot)\underline{e}_{2},\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{G}^{\pm}(z)(y,\cdot)\underline{e}_{2}\rangle\,\underline{e}_{2}\underline{e}_{2}^{\top}\\\
&=:e^{\pm iz(|x|+|y|)}A_{1}^{\pm}(z,x,y)+e^{\pm
iz|x|}A_{2}^{\pm}(z,x,y)+e^{\pm
iz|y|}A_{3}^{\pm}(z,x,y)+A_{4}^{\pm}(z,x,y).\end{split}$

We plug this identity into the left hand side of (6.22), and hence it will be
sufficient to provide the bounds

$\left|\int_{0}^{\infty}e^{itz^{2}\pm
izr}z\chi_{\mathrm{m}}(z^{2})A_{k}^{\pm}(z,x,y)\,\mathrm{d}z\right|\lesssim\min\\{|t|^{-\frac{1}{2}},|t|^{-\frac{3}{2}}\langle
r\rangle\\},\quad k\in\\{1,\ldots,4\\},$ (6.24)

where $r$ can represent $0$ or $|x|$, $|y|$, or the sum of both variables. For
the case $k=1$, by Lemma 2.1, we have that

$\left|\int_{0}^{\infty}e^{itz^{2}\pm
iz(|x|+|y|)}z\chi_{\mathrm{m}}(z^{2})A_{1}^{\pm}(z,x,y)\,\mathrm{d}z\right|\leq
C|t|^{-\frac{1}{2}}\|\partial_{z}\left(z\chi_{\mathrm{m}}(z^{2})A_{1}^{\pm}(z,x,y)\right)\|_{L_{z}^{1}({\mathbb{R}})}.$

Since the term $z\chi_{\mathrm{m}}(z^{2})$ is smooth and has compact support,
we only need to track the derivatives when they fall onto either
$\mathcal{G}^{\pm}(z)$ or $\mathcal{R}^{\pm}(z)$. In any case, thanks to the
exponential decay of $\mathcal{V}$, and the bounds (6.13), (6.14) from the
previous lemmas, we have the following uniform bound

$\begin{split}&\sup_{\pm}\sup_{z\in\operatorname*{supp}(\chi_{\mathrm{m}})}\sup_{j,k=1,2}|\partial_{z}\langle(\mathcal{G}^{\pm})^{*}(z)(y,\cdot)\underline{e}_{j},\mathcal{V}\mathcal{R}^{\pm}(z)\mathcal{V}\mathcal{G}^{\pm}(z)(x,\cdot)\underline{e}_{k}\rangle|\\\
&\lesssim\sup_{\pm}\sup_{z\in\operatorname*{supp}(\chi_{\mathrm{m}})}\sup_{j,k=1,2}\left\|\sqrt{|\mathcal{V}|}(x_{1})\left(|\mathcal{R}^{\pm}(z)(x_{1},x_{2})|+|\partial_{z}\mathcal{R}^{\pm}(z)(x_{1},x_{2})|\right)\sqrt{|\mathcal{V}|}(x_{2})\right\|_{L_{x_{2}}^{2}\to
L_{x_{1}}^{2}}\\\
&\quad\cdot\|\sqrt{|\mathcal{V}|}(x_{1})\left(|\mathcal{G}^{\pm}(z)(x,x_{1})|+|\partial_{z}\mathcal{G}^{\pm}(z)(x,x_{1})|\right)\underline{e}_{j}\|_{L_{x_{1}}^{2}}\\\
&\quad\cdot\|\sqrt{|\mathcal{V}|}(x_{2})\left(|\mathcal{G}^{\pm}(z)(x_{2},y)|+|\partial_{z}\mathcal{G}^{\pm}(z)(x_{2},y)|\right)\underline{e}_{k}\|_{L_{x_{2}}^{2}}\\\
&\lesssim 1,\end{split}$ (6.25)

for all $x,y\in{\mathbb{R}}$.

To prove the weighted dispersive estimate, we invoke the stronger estimate in
Lemma 2.1:

$\left|\int_{0}^{\infty}e^{itz^{2}\pm
iz(|x|+|y|)}z\chi_{\mathrm{m}}(z^{2})A_{1}^{\pm}(z,x,y)\,\mathrm{d}z\right|\leq
C|t|^{-\frac{3}{2}}\left\|[\partial_{z}^{2}\pm
i(|x|+|y|)\partial_{z}]\left(\chi_{\mathrm{m}}(z^{2})A_{1}^{\pm}(z,x,y)\right)\right\|_{L_{z}^{1}({\mathbb{R}})}$

Here, we can apply the same argument as in (6.25) for the two derivatives
bound on $A_{1}^{\pm}$ using the estimates (6.13) and (6.15), whereas the
bound on one derivative for $A_{1}^{\pm}$ leads to the weights $\langle
x\rangle\langle y\rangle$. Thus, we prove (6.24) for $k=1$. The other cases
follow by the same argument and we are done. ∎

Finally, we conclude with the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

By combining the estimates from Proposition 5.2, Proposition 6.1, and
Proposition 6.2, we have established the bounds

$\left\|e^{it\mathcal{H}}P_{\mathrm{s}}^{+}{\vec{u}}\,\right\|_{L_{x}^{\infty}({\mathbb{R}})\times
L_{x}^{\infty}({\mathbb{R}})}\lesssim|t|^{-\frac{1}{2}}\|{\vec{u}}\,\|_{L_{x}^{1}({\mathbb{R}})\times
L_{x}^{1}({\mathbb{R}})},$

as well as

$\left\|\langle
x\rangle^{-2}(e^{it\mathcal{H}}P_{\mathrm{s}}^{+}-F_{t}^{+}){\vec{u}}\,\right\|_{L_{x}^{\infty}({\mathbb{R}})\times
L_{x}^{\infty}({\mathbb{R}})}\lesssim|t|^{-\frac{3}{2}}\|{\vec{u}}\,\|_{L_{x}^{1}({\mathbb{R}})\times
L_{x}^{1}({\mathbb{R}})},$

for any
${\vec{u}}:=(u_{1},u_{2})^{\top}\in\mathcal{S}({\mathbb{R}})\times\mathcal{S}({\mathbb{R}})$
and $|t|\geq 1$, with $F_{t}^{+}$ given by (5.9). By Remark 3.3, we can
similarly deduce that the unweighted dispersive estimate for the evolution
$e^{it\mathcal{H}}P_{\mathrm{s}}^{-}$ using the identity (3.17). On the other
hand, for the weighted estimate, we find that the leading contribution to
$e^{it\mathcal{H}}P_{\mathrm{s}}^{-}$ is given by

$F_{t}^{-}(x,y)=\sigma_{1}F_{-t}^{+}(x,y)\sigma_{1}=-\frac{e^{-it\mu}}{\sqrt{4\pi
it}}[\sigma_{1}\Psi(x)][\sigma_{3}\sigma_{1}\Psi(y)]^{*},$ (6.26)

where we used the anti-commutation identity
$\sigma_{3}\sigma_{1}=-\sigma_{1}\sigma_{3}$. Thus, we conclude the local
decay estimate (1.14) and the formula (1.15) by setting
$F_{t}:=F_{t}^{+}+F_{t}^{-}$. ∎

##  Appendix A Neumann series

######  Lemma A.1.

Let $A$ be an invertible operator and $B$ be a bounded operator satisfying
$\|B\|<\|A^{-1}\|^{-1}$. Then, $A-B$ is invertible with

$(A-B)^{-1}=A^{-1}\sum_{n=0}^{\infty}(BA^{-1})^{n}=A^{-1}+A^{-1}BA^{-1}+A^{-1}BA^{-1}BA^{-1}+\cdots,$
(A.1)

and

$\|(A-B)^{-1}\|\leq(\|A^{-1}\|^{-1}-\|B\|)^{-1}.$ (A.2)

###### Proof.

By the hypothesis $\|B\|<\|A^{-1}\|^{-1}$, we have $\|A^{-1}B\|<1$. Consider
the identity

$(A-B)^{-1}=(I-A^{-1}B)^{-1}A^{-1}.$

The term on the right hand side can be written in the usual Neumann series

$(I-A^{-1}B)^{-1}=\sum_{n=0}^{\infty}(A^{-1}B)^{n}.$

Thus, by multiplying $A^{-1}$, we deduce (A.1). Note that the argument also
holds true for $(A-B)^{-1}=A^{-1}(I-BA^{-1})^{-1}$. Now, since we have the
estimate

$\|(I-A^{-1}B)^{-1}\|\leq(1-\|A^{-1}B\|)^{-1},$

we deduce (A.2) by the sub-multiplicative property for operator norms. ∎

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# The fundamental Lepage form in two independent variables: a generalization
using order-reducibility

Zbyněk Urban and Jana Volná  Z. Urban and J. Volná
Department of Mathematics, Faculty of Civil Engineering
VŠB-Technical University of Ostrava
Ludvíka Podéště 1875/17, 708 33 Ostrava-Poruba, Czech Republic
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A second-order generalization of the fundamental Lepage form of geometric
calculus of variations over fibered manifolds with $2$-dimensional base is
described by means of insisting on (i) equivalence relation “ _Lepage
differential $2$-form is closed if and only if the associated Lagrangian is
trivial_” and, (ii) the principal component of Lepage form, extending the
well-known Poincaré–Cartan form, preserves order prescribed by a given
Lagrangian. This approach completes several attempts of finding a Lepage
equivalent of a second-order Lagrangian possessing condition (i), which is
well-known for first-order Lagrangians in field theory due to Krupka and
Betounes.

###### Key words and phrases:

Lagrangian; Lepage equivalent; Poincaré-Cartan form; fundamental form;
calculus of variations; field theory; jet; fibered manifold.

###### 2000 Mathematics Subject Classification:

58A10; 58A20; 58E30; 70S05

##  1\. Introduction

Lepage forms play a basic role in the calculus of variations of both simple-
and multiple-integral problems over fibered manifolds and Grassmann
fibrations. Among the well-known examples of Lepage forms we mention namely:
the Cartan form of classical mechanics and its generalization in higher-order
mechanics (Krupka [11]); in first-order field theory, the Poincaré–Cartan form
(García [5]), the Carathéodory form (Carathéodory [2]), and the fundamental
Lepage form (also known as the Krupka–Betounes form) (Krupka [12], Betounes
[1]); in second-order field theory, the generalized Poincaré–Cartan form
(Krupka [13]), the generalized Carathéodory form (Crampin and Saunders [3],
Urban and Volná [26]), the fundamental Lepage form for second-order,
homogeneous Lagrangians (Saunders and Crampin [23]). See also Gotay [8],
Goldschimdt and Sternberg [7], Rund [21], Dedecker [4], Horák and Kolář [9],
Krupka [13], Krupka and Štěpánková [14], Saunders [22], Sniatycki [24].
Further recent attempts of generalization and study of the fundamental Lepage
equivalent for first- and second-order Lagrangians include also [19, 20, 25].
For a review of basic properties and results, see Krupka, Krupková, and
Saunders [17].

Replacing the initial Lagrangian by its Lepage equivalent, the corresponding
variational functional is preserved, and in addition the basic variational
properties as variations, extremals, or conservation laws can be formulated
and studied using geometric operations (such as the exterior derivative, the
Lie derivative) acting on the corresponding Lepage equivalent of a Lagrangian.

Our aim in this note is to study a generalization of the fundamental Lepage
equivalent $Z_{\lambda}$ of a  _second-order_ Lagrangian $\lambda$. This
Lepage equivalent was introduced for _first-order_ Lagrangians in variational
theory over fibered manifolds with an $n$-dimensional base (see [12, 1]), and
it obeys the following crucial property:

(1.1) $dZ_{\lambda}=0\quad\mathrm{if\,and\,only\,if}\quad E_{\lambda}=0,$

that is, the Lepage equivalent of a Lagrangian is closed if and only if the
Lagrangian is trivial (i.e., the corresponding Euler–Lagrange expressions
vanish identically).

For $2$-dimensional base (i.e., two independent variables), we show that a
Lepage equivalent $Z_{\lambda}$ of a second-order Lagrangian $\lambda$,
obeying the aforementioned equivalence property, does _not_ exist in general
when $Z_{\lambda}=\Theta_{\lambda}$ _plus_ a 2-contact part, where
$\Theta_{\lambda}$ is the principal component (Lepage form) of $Z_{\lambda}$.
Nevertheless, we describe here the fundamental Lepage form, associated with a
second-order Lagrangian which assures that the principal component of a Lepage
form (the generalized Poincaré–Cartan form) has the _same order_ as the
initial Lagrangian. This order reducibility assumption is also motivated by
the first-order theory and include, for instance, important class of
Lagrangians _linear_ in second derivatives.

Recent studies on the fundamental Lepage form include namely Saunders and
Crampin [23] (for two independent variables, generalization of the fundamental
form is given for higher-order, homogeneous Largangians on tangent bundles),
and Palese, Rossi, and Zanello [19] (on the basis of integration by parts,
possible generalization of the fundamental form for a second-order Lagrangian
is discussed which, however, differs from our result for $n=2$).

Basic underlying geometric structures, well adapted to the present paper, can
be found in book chapters Volná and Urban [28], and Krupka [16]. Throughout,
we use the standard geometric concepts: the exterior derivative $d$, the
contraction $i_{\xi}\rho$ of a differential form $\rho$ with respect to a
vector field $\xi$, and the pull-back operation $*$ acting on differential
forms.

If $(U,\varphi)$, $\varphi=(x^{i})$, is a chart on smooth manifold $X$, the
local volume element is denoted by
${\color[rgb]{0,0,0}\omega_{0}}=dx^{1}\wedge\ldots\wedge dx^{n}$, and we put

${\color[rgb]{0,0,0}\omega_{j}}{\color[rgb]{0,0,0}=i_{\partial/\partial
x^{j}}\omega_{0}=\frac{1}{(n-1)!}\varepsilon_{ji_{2}\ldots
i_{n}}dx^{i_{2}}\wedge\ldots\wedge dx^{i_{n}},}$

where $\varepsilon_{i_{1}i_{2}\ldots i_{n}}$ is the Levi-Civita permutation
symbol. We denote by $Y$ a fibered manifold of dimension $n+m$ over an
$n$-dimensional base manifold $X$ with projection $\pi:Y\rightarrow X$ the
surjective submersion. $J^{r}Y$ denotes the $r$-th order jet prolongation of
$Y$ whose elements are $r$-jets $J_{x}^{r}\gamma$ of sections $\gamma$ of
$\pi$ with source at $x\in X$ and target at $\gamma(x)\in Y$. The canonical
jet bundle projection is denoted by $\pi^{r,s}:J^{r}Y\rightarrow J^{s}Y$.
Every fibered chart $(V,\psi)$, $\psi=(x^{i},y^{\sigma})$, $1\leq i\leq n$,
$1\leq\sigma\leq m$,on $Y$ induces the associated chart
$(U,\varphi),\varphi=(x^{i})$ on $X$, and the associated fibered chart
$(V^{r},\psi^{r})$ on $J^{r}Y$, where $U=\pi(V)$, $V^{r}=(\pi^{r,0})^{-1}(V)$,
and $\psi^{r}=(x^{i},y^{\sigma},y_{j}^{\sigma},\ldots,y_{j_{1}\ldots
j_{r}}^{\sigma})$, where

$y_{j_{1}\ldots j_{k}}^{\sigma}(J_{x}^{r}\gamma)=D_{j_{1}}\ldots
D_{j_{k}}(y^{\sigma}\gamma\varphi^{-1})(\varphi(x)),\quad 0\leq k\leq r.$

A tangent vector $\xi\in T_{y}Y$ is called $\pi$_-vertical_ , if
$T\pi\cdot\xi=0$, and a differential form $\rho$ on $Y$ is called
$\pi$-_horizontal_ , if for every point $y\in Y$ the contraction
$i_{\xi}\rho(y)$ vanishes whenever $\xi\in T_{y}Y$ is $\pi$-vertical.

We denote by $\Omega_{q}^{r}Y$ the $\Omega_{0}^{r}Y$-module of smooth
differential $q$-forms defined on $J^{r}Y$. $\pi^{r}$-horizontal $q$-forms on
$J^{r}Y$ constitute a submodule of $\Omega_{q}^{r}Y$, denoted by
$\Omega_{q,X}^{r}Y$. For a fibered manifold $\pi:Y\rightarrow X$ there exists
a unique morphism $h:\Omega^{r}Y\rightarrow\Omega^{r+1}Y$ of exterior algebras
of differential forms such that for any fibered chart $(V,\psi)$,
$\psi=(x^{i},y^{\sigma})$, on $Y$, and any differentiable function
$f:J^{r}Y\rightarrow\mathbb{R}$,

$hf=f\circ\pi^{r+1,r},\quad\quad hdf=(d_{i}f)dx^{i},$

where $d_{i}$ (resp. $d_{i}^{\prime}$) is the $i$-th formal derivative (resp.
the cut $i$-th formal derivative) operator associated with $(V,\psi)$,

$d_{i}=d_{i}^{\prime}+\sum_{j_{1}\leq\ldots\leq j_{r}}\frac{\partial}{\partial
y_{j_{1}\ldots j_{r}}^{\sigma}}y_{j_{1}\ldots j_{r}i}^{\sigma},$

and

(1.2) $d_{i}^{\prime}=\frac{\partial}{\partial
x^{i}}+\sum_{k=0}^{r-1}\sum_{j_{1}\leq\ldots\leq
j_{k}}\frac{\partial}{\partial y_{j_{1}\ldots j_{k}}^{\sigma}}y_{j_{1}\ldots
j_{k}i}^{\sigma}.$

A differential form $q$-form $\rho\in\Omega_{q}^{r}Y$ satisfying $h\rho=0$ is
called _contact_ , and every contact form $\rho$ is generated by contact
$1$-forms

$\omega_{j_{1}\ldots j_{s}}^{\sigma}=dy_{j_{1}\ldots
j_{s}}^{\sigma}-y_{j_{1}\ldots j_{s}i}^{\sigma}dx^{i},\qquad 0\leq s\leq r-1.$

Any differential $q$-form $\rho\in\Omega_{q}^{r}Y$ has a unique invariant
decomposition,

$(\pi^{r+1,r})^{*}\rho=h\rho+\sum_{k=1}^{q}p_{k}\rho,$

where $p_{k}\rho$ is the $k$-_contact component_ of $\rho$, containing exactly
$k$ exterior product factors $\omega_{j_{1}\ldots j_{s}}^{\sigma}$ with
respect to any fibered chart $(V,\psi)$.

##  2\. Lepage equivalents in field theory

We summarize basic facts about Lepage differential forms on finite-order jet
prolongations of fibered manifolds and, in particular, we discuss
distinguished examples of Lepage equivalents of first- and second-order
Lagrangians; for more details see [13, 16, 18, 28, 25, 26].

By a Lagrangian $\lambda$ for a fibered manifold $\pi:Y\rightarrow X$ of order
$r$ we mean an element of the submodule $\Omega_{n,X}^{r}Y$ of
$\pi^{r}$-horizontal $n$-forms in the module of $n$-forms $\Omega_{n}^{r}Y$,
defined on the $r$-th jet prolongation $J^{r}Y$. In a fibered chart
$(V,\psi)$, $\psi=(x^{i},y^{\sigma})$, Lagrangian
$\lambda\in\Omega_{n,X}^{r}Y$ has an expression

(2.1) $\lambda=\mathscr{L}\omega_{0},$

where $\omega_{0}=dx^{1}\wedge\ldots\wedge dx^{n}$ is the (local) volume
element, and $\mathscr{L}:V^{r}\rightarrow\mathbb{R}$ is said to be the
Lagrange function associated to $\lambda$ and $(V,\psi)$.

An $n$-form $\rho\in\Omega_{n}^{s}Y$ on $J^{s}Y$ is called a Lepage form, if
one of the following equivalent conditions is satisfied:

(i) $p_{1}d\rho$ is a $\pi^{s+1,0}$-horizontal $(n+1)$-form,

(ii) $hi_{\xi}d\rho=0$ for arbitrary $\pi^{s,0}$-vertical vector field $\xi$
on $J^{s}Y$,

(iii) For every fibered chart $(V,\psi)$ on $Y$, $\rho$ satisfies

$(\pi^{s+1,s})^{*}\rho=f_{0}\omega_{0}+\sum_{k=0}^{s}f_{\sigma}^{i,j_{1}\ldots
j_{k}}\omega_{j_{1}\ldots j_{k}}^{\sigma}\wedge\omega_{i}+\eta,$

where $n$-form $\eta$ has order of contactness $\geq 2$, and

$\displaystyle\frac{\partial f_{0}}{\partial y_{j_{1}\ldots
j_{k}}^{\sigma}}-d_{i}f_{\sigma}^{i,j_{1}\ldots
j_{k}}-f_{\sigma}^{j_{k},j_{1}\ldots j_{k-1}}$
$\displaystyle=0\qquad\mathrm{Sym}(j_{1}\ldots j_{k}),\,\,k\leq s,$
$\displaystyle\frac{\partial f_{0}}{\partial y_{j_{1}\ldots
j_{s+1}}^{\sigma}}-f_{\sigma}^{j_{s+1},j_{1}\ldots j_{s}}$
$\displaystyle=0\qquad\mathrm{Sym}(j_{1}\ldots j_{s+1}).$

Let $\lambda\in\Omega_{n,X}^{r}Y$ be a Lagrangian for $\pi:Y\rightarrow X$. A
Lepage form $\rho\in\Omega_{n}^{s}Y$ is called a  _Lepage equivalent_ of
$\lambda$, if $h\rho=\lambda$ (up to a canonical jet projection). The
following theorem describes the structure of Lepage equivalents of a
Lagrangian.

######  Theorem 1.

Let $\lambda\in\Omega_{n,X}^{r}Y$ be a Lagrangian of order $r$ for
$\pi:Y\rightarrow X$, locally expressed by (2.1) with respect to a fibered
chart $(V,\psi)$. An $n$-form $\rho\in\Omega_{n}^{s}Y$ is a Lepage equivalent
of $\lambda$ if and only if it obeys the following decomposition,

(2.2) $(\pi^{s+1,s})^{*}\rho=\Theta_{\lambda}+d\mu+\eta,$

where $n$-form $\Theta_{\lambda}$ is defined on $V^{2r-1}$ by

(2.3)
$\Theta_{\lambda}=\mathscr{L}\omega_{0}+\sum_{k=0}^{r-1}\left(\sum_{l=0}^{r-1-k}(-1)^{l}d_{p_{1}}\ldots
d_{p_{l}}\frac{\partial\mathscr{L}}{\partial y_{j_{1}\ldots j_{k}p_{1}\ldots
p_{l}i}^{\sigma}}\right)\omega_{j_{1}\ldots j_{k}}^{\sigma}\wedge\omega_{i},$

$\mu$ is a contact $(n-1)$-form, and an $n$-form $\eta$ has the order of
contactness $\geq 2$.

###### Proof.

See [18, 28]. ∎

$\Theta_{\lambda}$, given by (2.3) on $V^{2r-1}$, is called the _principal_
_Lepage equivalent_ of $\lambda$ with respect to fibered chart $(V,\psi)$.
This Lepage form is uniquelly determined by imposing that a Lepage form is
$\pi^{2r-1,r-1}$-horizontal and it has the order of contactness $\leq 1$._We
n_ ote that, in general, decomposition (2.2) is _not_ uniquely determined with
respect to contact forms $\mu$ and $\eta$, although the Lepage equivalent
$\rho$ satisfying (2.2) is a globally defined differential form on $J^{s}Y$.
For a first-order Lagrangian $\lambda$, $\Theta_{\lambda}$ (2.3) is the well-
known _Poincaré–Cartan form_ defined on $J^{1}Y$ (cf. [5]),

(2.4)
$\Theta_{\lambda}=\mathscr{L}\omega_{0}+\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}\omega^{\sigma}\wedge\omega_{j}.$

For a second-order Lagrangian $\lambda$, $\Theta_{\lambda}$ (2.3) is the
_generalized Poincaré–Cartan form_ defined on $J^{3}Y$ (cf. [13, 16]),

(2.5)
$\Theta_{\lambda}=\mathscr{L}\omega_{0}+\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\omega^{\sigma}\wedge\omega_{j}+\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\omega_{i}^{\sigma}\wedge\omega_{j}.$

We point out that for Lagrangians of order $r\geq 3$, local expressions (2.3)
need _not_ define differential forms on $J^{2r-1}Y$ globally (cf. [9, 13]).

The well-known _Euler–Lagrange mapping_ of the calculus of variations assigns
to a Lagrangian $\lambda\in\Omega_{n,X}^{r}Y$ the _Euler–Lagrange form_

(2.6) $E_{\lambda}=E_{\sigma}(\mathscr{L})\omega^{\sigma}\wedge\omega_{0},$

with coefficients

(2.7) $E_{\sigma}(\mathscr{L})=\sum_{k=0}^{r}(-1)^{k}d_{i_{1}}\ldots
d_{i_{k}}\frac{\partial\mathscr{L}}{\partial y_{i_{1}\ldots i_{k}}^{\sigma}}$

the _Euler–Lagrange expressions_ associated to
$\mathscr{L}:V^{r}\rightarrow\mathbb{R}$. Note that the $1$-contact and
$\pi^{r,0}$-horizontal $(n+1)$-form $E_{\lambda}$ is defined by means of
(2.6), (2.7) on $J^{2r}Y$ globally. The following theorem explains a crucial
relation between a Lepage equivalent of Lagrangian $\lambda$ on one side and
the associated Euler–Lagrange form $E_{\lambda}$ on the other side.

######  Theorem 2.

Let $\lambda\in\Omega_{n,X}^{r}Y$ be a Lagrangian of order $r$ for
$\pi:Y\rightarrow X$ and let $\rho\in\Omega_{n}^{s}Y$ be a Lepage equivalent
of $\lambda$. Then

$(\pi^{s+1,s})^{*}d\rho=E_{\lambda}+F,$

where $E_{\lambda}$ is the Euler–Lagrange form (2.6) associated to $\lambda$,
and $F$ is an $(n+1)$-form with order of contactness $\geq 2$. In particular ,
$E_{\lambda}$ coincides with the $1$-contact component of the exterior
derivative of a Lepage equivalent of Lagrangian $\lambda$, i.e. on $J^{2r}Y$,

(2.8) $E_{\lambda}=p_{1}d\rho.$

###### Proof.

See [16, 18, 28]. ∎

Beside the principal Lepage equivalent $\Theta_{\lambda}$, given by (2.4) and
(2.5) for a first- and second-order Lagrangian $\lambda$, respectively, we
recall the other known examples of Lepage equivalents, determined by means of
additional requirements.

######  Lemma 3.

_(a)_ Let $\lambda\in\Omega_{n,X}^{1}Y$ be a non-vanishing first-order
Lagrangian for $\pi:Y\rightarrow X$, locally expressed by (2.1). Then the
local expression

(2.9) $\displaystyle\Lambda_{\lambda}$
$\displaystyle=\frac{1}{\mathscr{L}^{n-1}}\bigwedge_{j=1}^{n}\left(\mathscr{L}dx^{j}+\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}\omega^{\sigma}\right)$

defines a $\pi^{1,0}$-horizontal differential $n$-form
$\Lambda_{\lambda}\in\Omega_{n}^{1}Y$, which is a Lepage equivalent of
$\lambda$.

_(b)_ Let $\lambda\in\Omega_{n,X}^{2}Y$ be a non-vanishing second-order
Lagrangian for $\pi:Y\rightarrow X$, locally expressed by (2.1). Then the
local expression

(2.10) $\displaystyle\Lambda_{\lambda}$
$\displaystyle=\frac{1}{\mathscr{L}^{n-1}}\bigwedge_{j=1}^{n}\left(\mathscr{L}dx^{j}+\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\omega^{\sigma}+\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\omega_{i}^{\sigma}\right)$

defines a $\pi^{3,1}$-horizontal differential $n$-form
$\Lambda_{\lambda}\in\Omega_{n}^{3}Y$, which is a Lepage equivalent of
$\lambda$.

###### Proof.

See [2, 26]. ∎

$\Lambda_{\lambda}$ (2.9) is the well-known _Carathéodory form,_ associated to
a non-vanishing, first-order Lagrangian _ $\lambda$ _(cf. [2])_,_ whereas __
$\Lambda_{\lambda}$ (2.10) is its generalization for second-order Lagrangians,
recently studied in [26].

######  Remark 4.

Note that the Carathéodory form $\Lambda_{\lambda}$ (2.10) is decomposed as a
sum of the generalized Poincaré–Cartan form $\Theta_{\lambda}$ (2.5) and a
$\pi^{3,1}$-horizontal, $2$-contact differential $n$-form. For further
purpose, we give this decomposition explicitly for the dimension of base
$n=2$:

$\displaystyle\Lambda_{\lambda}$ $\displaystyle=\Theta_{\lambda}$
$\displaystyle+\frac{1}{\mathscr{L}}\left(\frac{\partial\mathscr{L}}{\partial
y_{1}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{i1}^{\sigma}}\right)\left(\frac{\partial\mathscr{L}}{\partial
y_{2}^{\nu}}-d_{k}\frac{\partial\mathscr{L}}{\partial
y_{k2}^{\nu}}\right)\omega^{\sigma}\wedge\omega^{\nu}$ (2.11)
$\displaystyle+\frac{1}{\mathscr{L}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j2}^{\nu}}\left(\frac{\partial\mathscr{L}}{\partial
y_{1}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{i1}^{\sigma}}\right)-\frac{\partial\mathscr{L}}{\partial
y_{j1}^{\nu}}\left(\frac{\partial\mathscr{L}}{\partial
y_{2}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{i2}^{\sigma}}\right)\right)\omega^{\sigma}\wedge\omega_{j}^{\nu}$
$\displaystyle+\frac{1}{\mathscr{L}}\frac{\partial\mathscr{L}}{\partial
y_{i1}^{\sigma}}\frac{\partial\mathscr{L}}{\partial
y_{j2}^{\nu}}\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}.$

######  Lemma 5.

Let $\lambda\in\Omega_{n,X}^{1}Y$ be a first-order Lagrangian for
$\pi:Y\rightarrow X$, locally expressed by (2.1). There exists a unique Lepage
equivalent $Z_{\lambda}\in\Omega_{n}^{1}Y$ of $\lambda$, which satisfies
$Z_{\lambda}=(\pi^{1,0})^{*}\rho$ for any $n$-form $\rho\in\Omega_{n}^{0}W$ on
$W$ such that $h\rho=\lambda$. With respect to a fibered chart $(V,\psi)$,
$Z_{\lambda}$ has an expression

(2.12) $\displaystyle Z_{\lambda}$
$\displaystyle=\mathscr{L}\omega_{0}+\sum_{k=1}^{n}\frac{1}{(n-k)!}\frac{1}{(k!)^{2}}\frac{\partial^{k}\mathscr{L}}{\partial
y_{j_{1}}^{\sigma_{1}}\ldots\partial
y_{j_{k}}^{\sigma_{k}}}\varepsilon_{j_{1}\ldots j_{k}i_{k+1}\ldots i_{n}}$
$\displaystyle\quad\cdot\omega^{\sigma_{1}}\land\ldots\wedge\omega^{\sigma_{k}}\wedge
dx^{i_{k+1}}\wedge\ldots\wedge dx^{i_{n}}.$

###### Proof.

See [12, 1]. ∎

$Z_{\lambda}$ (2.12) is known as the _fundamental Lepage form,_ or
_Krupka–Betounes form,_ associated to first-order Lagrangian _$\lambda$ ;_
original sources are [12] and [1], further recent contributions include [19,
20, 25].

######  Remark 6.

One can directly verify that the Lepage equivalent $Z_{\lambda}$ (2.12)
satisfies the equivalence relation “$Z_{\lambda}$_is closed if and only if the
associated Lagrangian $\lambda=hZ_{\lambda}$ is trivial_”, that is
“$dZ_{\lambda}=0$ if and only if $E_{\lambda}=0$.” Since
$E_{\lambda}=p_{1}dZ_{\lambda}$ (2.8), it is immediate that
$\lambda\in\Omega_{n,X}^{1}Y$ is trivial, provided $Z_{\lambda}$_is closed.
However, the converse implication if a non-trivial one._ A construction of
(local, 1-contact) Lepage equivalents of higher-order Lagrangians satisfying
this equivalence relation has been recently described in [27].

Another remarkable property which $Z_{\lambda}$ (2.12) satisfies is the
following: $Z_{\lambda}$ is $\pi^{1,0}$-projectable if and only $E_{\lambda}$
is $\pi^{2,1}$-projectable.

##  3\. Trivial Lagrangians

A Lagrangian $\lambda$ is said to be _variationally trivial_ (or _null_), if
the associated Euler–Lagrange form $E_{\lambda}$ vanishes identically. The
following theorem describes variationally trivial Lagrangians.

######  Theorem 7.

Let $\lambda\in\Omega_{n,X}^{r}Y$ be a Lagrangian of order $r$ for
$\pi:Y\rightarrow X$. The following conditions are equivalent:

_(i)_ $\lambda$ is trivial.

_(ii)_ For every fibered chart $(V,\psi)$ on $Y$ there exists an $(n-1)$-form
$\mu\in\Omega_{n-1}^{r-1}Y$ such that on $V^{r}$,

$\lambda=hd\mu.$

_(iii)_ For every fibered chart $(V,\psi)$ on $Y$ there exist functions
$g^{i}:V^{r}\rightarrow\mathbb{R}$ such that $\lambda=\mathscr{L}\omega_{0}$
on $V^{r}$, where

(3.1) $\mathscr{L}=d_{i}g^{i}.$

###### Proof.

See [15]; also cf. [28]. ∎

As a consequence of Theorem 7, we now summarize explicit chart conditions for
a trivial Lagrangian of _second-order_ , needed later in this paper. Let
$\lambda\in\Omega_{n,X}^{2}Y$ be a Lagrangian for $Y$, locally expressed by
$\lambda=\mathscr{L}\omega_{0}$ (2.1) with respect to a fibered chart
$(V,\psi)$, $\psi=(x^{i},y^{\sigma})$, on $Y$, where
$\mathscr{L}=\mathscr{L}(x^{i},y^{\sigma},y_{j}^{\sigma},y_{jk}^{\sigma})$,
$j\leq k$, is the associated Lagrange function defined on $V^{2}\subset
J^{2}Y$, and $E_{\sigma}(\mathscr{L})$ are the corresponding Euler–Lagrange
expressions on $V^{4}\subset J^{4}Y$,

(3.2) $E_{\sigma}(\mathscr{L})=\frac{\partial\mathscr{L}}{\partial
y^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{i}^{\sigma}}+d_{i}d_{j}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}.$

######  Lemma 8.

_(a)_ $\lambda$ is trivial if and only if it satisfies the following
conditions

(3.3) $\displaystyle\frac{\partial\mathscr{L}}{\partial
y^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{i}^{\sigma}}+d_{i}^{\prime}d_{j}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}=0,$ (3.4)
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y_{p}^{\nu}\partial
y_{iq}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial y_{pq}^{\nu}\partial
y_{i}^{\sigma}}+2d_{j}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{ij}^{\sigma}}\right)=0\qquad\mathrm{Sym}(pqi),$ (3.5)
$\displaystyle\frac{\partial^{3}\mathscr{L}}{\partial y_{st}^{\mu}\partial
y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}=0\qquad\mathrm{Sym}(stj),\,\,\mathrm{Sym}(pqi),$ (3.6)
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}=0\qquad\mathrm{Sym}(pqij),$

where $d_{i}^{\prime}$ is the cut formal derivative operator _( 1.2)._

_(b)_ For every fibered chart $(V,\psi)$ on $Y$ there exist functions
$g^{i}:V^{2}\rightarrow\mathbb{R}$ such that $\lambda=\mathscr{L}\omega_{0}$
on $V^{2}$, where

$\mathscr{L}=d_{i}g^{i},$

and

(3.7) $\frac{\partial g^{i}}{\partial y_{jk}^{\sigma}}+\frac{\partial
g^{j}}{\partial y_{ki}^{\sigma}}+\frac{\partial g^{k}}{\partial
y_{ij}^{\sigma}}=0.$

###### Proof.

Consider the Euler–Lagrange expressions $E_{\sigma}(\mathscr{L})$, associated
to a second-order Lagrangian $\lambda=\mathscr{L}\omega_{0}$. We have

$\displaystyle d_{i}\frac{\partial\mathscr{L}}{\partial y_{i}^{\sigma}}$
$\displaystyle=\frac{\partial^{2}\mathscr{L}}{\partial x^{i}\partial
y_{i}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial y^{\nu}\partial
y_{i}^{\sigma}}y_{i}^{\nu}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{p}^{\nu}\partial
y_{i}^{\sigma}}y_{pi}^{\nu}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{i}^{\sigma}}y_{pqi}^{\nu}$
$\displaystyle=d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{i}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial y_{pq}^{\nu}\partial
y_{i}^{\sigma}}y_{pqi}^{\nu},$

and

$\displaystyle d_{i}d_{j}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}=d_{i}\left(d_{j}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}y_{pqj}^{\nu}\right)$
$\displaystyle=d_{i}^{\prime}d_{j}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}+d_{j}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}\right)y_{pqi}^{\nu}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{p}^{\nu}\partial y_{iq}^{\sigma}}y_{pqi}^{\nu}$
$\displaystyle+d_{i}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}\right)y_{pqj}^{\nu}+\frac{\partial^{3}\mathscr{L}}{\partial
y_{st}^{\mu}\partial y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}y_{sti}^{\mu}y_{pqj}^{\nu}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{ij}^{\sigma}}y_{pqji}^{\nu}.$

Thus (3.2) now reads,

$\displaystyle E_{\sigma}(\mathscr{L})$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial
y^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{i}^{\sigma}}+d_{i}^{\prime}d_{j}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}$ (3.8)
$\displaystyle+\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{p}^{\nu}\partial y_{iq}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial
y_{i}^{\sigma}}+2d_{j}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{ij}^{\sigma}}\right)\right)y_{pqi}^{\nu}$
$\displaystyle+\frac{\partial^{3}\mathscr{L}}{\partial y_{st}^{\mu}\partial
y_{pq}^{\nu}\partial
y_{ij}^{\sigma}}y_{sti}^{\mu}y_{pqj}^{\nu}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{ij}^{\sigma}}y_{pqij}^{\nu},$

hence $E_{\sigma}(\mathscr{L})$ vanish if and only if conditions (3.4) hold,
proving (a).

Condition (b) follows from Theorem 7, (iii), where (3.7) is implied by
equation (3.1) satisfied identically on $V^{2}\subset J^{2}Y$. ∎

##  4\. The fundamental Lepage equivalent of a second-order Lagrangian: order
reduction

First, we present an order reduction condition, which allows us to construct a
generalization of the fundamental Lepage equivalent $Z_{\lambda}$ (2.12) for a
second-order Lagrangian $\lambda$. From the Theorem 1 on the structure of
every Lepage equivalent of a Lagrangian, it is immediate that a differential
$n$-form to be found must be decomposable as
$Z_{\lambda}=\Theta_{\lambda}+d\mu+\eta$ (up to a canonical jet projection),
where $\Theta_{\lambda}$ is the generalized Poincaré–Cartan form (2.5), $\mu$
is a contact $(n-1)$-form, and an $n$-form $\eta$ has the order of contactness
$\geq 2$.

Let us assume that $\Theta_{\lambda}$ (2.5) is of _the same order_ as the
Lagrangian $\lambda$.

Note that this condition is automatically satisfied for first-order
Lagrangians but, nevertheless, for the second-order it restricts the class of
Lagrangians under consideration.

######  Lemma 9.

Let $\lambda\in\Omega_{n,X}^{2}Y$ be a second-order Lagrangian for
$\pi:Y\rightarrow X$. Then the generalized Poincaré–Cartan form
$\Theta_{\lambda}$ (2.5) is of second-order, if and only if for every chart
$(V,\psi)$, $\psi=(x^{i},y^{\sigma})$, on $Y$,

(4.1) $\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{ij}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{ip}^{\nu}\partial y_{qj}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{qi}^{\nu}\partial y_{pj}^{\sigma}}=0,$

where $\mathscr{L}$ is the Lagrange function, associated to $\lambda$ (2.1).

For $n=2$, (4.1) read

$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y_{11}^{\nu}\partial
y_{11}^{\sigma}}=0,\,\,\frac{\partial^{2}\mathscr{L}}{\partial
y_{11}^{\nu}\partial
y_{12}^{\sigma}}=0,\,\,\frac{\partial^{2}\mathscr{L}}{\partial
y_{22}^{\nu}\partial
y_{21}^{\sigma}}=0,\,\,\frac{\partial^{2}\mathscr{L}}{\partial
y_{22}^{\nu}\partial y_{22}^{\sigma}}=0,$ (4.2)
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y_{11}^{\nu}\partial
y_{22}^{\sigma}}+2\frac{\partial^{2}\mathscr{L}}{\partial y_{12}^{\nu}\partial
y_{12}^{\sigma}}=0.$

###### Proof.

The necessary and sufficient condition (4.1) follows immediately from the
chart expression (2.5) of $\Theta_{\lambda}$, by means of annihilating terms
linear in coordinates $y_{pqi}^{\tau}$. ∎

######  Lemma 10.

Let $\lambda\in\Omega_{n,X}^{2}Y$ be a second-order Lagrangian for
$\pi:Y\rightarrow X$ such that the generalized Poincaré–Cartan form
$\Theta_{\lambda}$ (2.5) is of second-order. Then $\lambda$ is variationally
trivial, if and only if for every chart $(V,\psi)$, $\psi=(x^{i},y^{\sigma})$,
on $Y$,

(4.3) $\displaystyle\frac{\partial\mathscr{L}}{\partial
y^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{i}^{\sigma}}+d_{i}^{\prime}d_{j}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}=0,\quad\frac{\partial^{2}\mathscr{L}}{\partial
y_{p}^{\nu}\partial y_{iq}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{pq}^{\nu}\partial y_{i}^{\sigma}}=0\qquad\mathrm{Sym}(pqi).$

For $n=2$, (4.3) read

$\displaystyle\frac{\partial\mathscr{L}}{\partial
y^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{i}^{\sigma}}+d_{i}^{\prime}d_{j}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}=0,\quad i,j=1,2,$
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y_{1}^{\nu}\partial
y_{11}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial y_{11}^{\nu}\partial
y_{1}^{\sigma}}=0,\quad\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial y_{22}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{22}^{\nu}\partial y_{2}^{\sigma}}=0,$ (4.4) $\displaystyle
2\frac{\partial^{2}\mathscr{L}}{\partial y_{1}^{\nu}\partial
y_{12}^{\sigma}}-2\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial y_{12}^{\nu}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial y_{11}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\sigma}\partial y_{11}^{\nu}}=0,$ $\displaystyle
2\frac{\partial^{2}\mathscr{L}}{\partial y_{2}^{\nu}\partial
y_{12}^{\sigma}}-2\frac{\partial^{2}\mathscr{L}}{\partial y_{12}^{\nu}\partial
y_{2}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial y_{1}^{\nu}\partial
y_{22}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial y_{22}^{\nu}\partial
y_{1}^{\sigma}}=0.$

###### Proof.

Necessary and sufficient conditions (4.3) for a variationally trivial
Lagrangian are nothing but reduction of (3.3)–(3.6), Lemma 8, with the help of
(4.1). ∎

In the next Theorem we construct the fundamental Lepage equivalent
$Z_{\lambda}$ of a second-order Lagrangian $\lambda$ over fibered manifolds
$\pi:Y\rightarrow X$, where $\dim X=2$.

Analogously to the Carathéodory form $\Lambda_{\lambda}$ (2.11), associated to
second-order Lagrangian $\lambda\in\Omega_{2,X}^{2}Y$, suppose that
$Z_{\lambda}$ is decomposed as a sum of the generalized Poincaré–Cartan form
$\Theta_{\lambda}$ (2.5) and contact terms, generated by the wedge products
$\omega^{\sigma}\wedge\omega^{\nu}$, $\omega^{\sigma}\wedge\omega_{j}^{\nu}$,
and $\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}$, i.e.

(4.5) $\displaystyle
Z_{\lambda}=\Theta_{\lambda}+\frac{1}{2}P_{\sigma\nu}\omega^{\sigma}\wedge\omega^{\nu}+Q_{\sigma,\nu}^{j}\omega^{\sigma}\wedge\omega_{j}^{\nu}+\frac{1}{2}R_{\sigma,\nu}^{i,j}\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu},$

where $P_{\sigma\nu}$, $Q_{\sigma,\nu}^{j}$, and $R_{\sigma,\nu}^{i,j}$ are
real-valued functions on $V^{3}\subset J^{3}Y$ such that $P_{\sigma\nu}$ is
skew-symmetric in $\left(\sigma,\nu\right)$, and $R_{\sigma,\nu}^{i,j}$ is
skew-symmetric in pairs $\left(i,\sigma\right),\left(j,\nu\right)$.

######  Theorem 11.

Let $\lambda\in\Omega_{2,X}^{2}Y$ be a second-order Lagrangian for
$\pi:Y\rightarrow X$ such that (4.1) holds. The following two conditions are
equivalent:

_(i)_ If $\lambda$ is variationally trivial, then $Z_{\lambda}$ (4.5) is
closed.

_(ii)_ For every chart $(V,\psi)$, $\psi=(x^{i},y^{\sigma})$, on $Y$,
$Z_{\lambda}$ (4.5) is uniquelly determined by means of real-valued functions
$P_{\sigma\nu}$, $Q_{\sigma,\nu}^{j}$, and $R_{\sigma,\nu}^{i,j}$, defined on
$V^{2}\subset J^{2}Y$ as

$\displaystyle P_{\sigma\nu}$
$\displaystyle=\frac{1}{2}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial y_{2}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{2}^{\sigma}}\right)$
$\displaystyle+d_{1}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{12}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial
y_{12}^{\nu}}\right)+d_{2}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\sigma}\partial y_{12}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial y_{12}^{\sigma}}\right),$ (4.6) $\displaystyle
Q_{\sigma,\nu}^{1}$ $\displaystyle=2\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial y_{12}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{12}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial
y_{11}^{\sigma}}-2d_{2}^{\prime}\frac{\partial^{2}\mathscr{L}}{\partial
y_{12}^{\sigma}\partial y_{12}^{\nu}},$ $\displaystyle Q_{\sigma,\nu}^{2}$
$\displaystyle=-2\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\sigma}\partial y_{12}^{\nu}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{22}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial
y_{12}^{\sigma}}+2d_{1}^{\prime}\frac{\partial^{2}\mathscr{L}}{\partial
y_{12}^{\sigma}\partial y_{12}^{\nu}},$ $\displaystyle R_{\sigma,\nu}^{1,2}$
$\displaystyle=-2\frac{\partial^{2}\mathscr{L}}{\partial
y_{12}^{\sigma}\partial y_{12}^{\nu}}=-R_{\sigma,\nu}^{2,1},$ $\displaystyle
R_{\sigma,\nu}^{1,1}$ $\displaystyle=0,\quad R_{\sigma,\nu}^{2,2}=0,$

###### Proof.

Suppose that Lagrangian $\lambda\in\Omega_{2,X}^{2}Y$ is variationally trivial
and the generalized Poincaré–Cartan form $\Theta_{\lambda}$ (2.5) is defined
on $J^{2}Y$. Thus, in abritrary fibered chart $(V,\psi)$,
$\psi=(x^{i},y^{\sigma})$, on $Y$, the associated Lagrange function
$\mathscr{L}:V^{2}\rightarrow\mathbb{R}$ satisfies conditions (3.3)–(3.5) of
Lemma 8, and (4.1) of Lemma 9. Note first that condition (4.1) already implies
(3.6), and using (4.1),

$d_{i}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}=d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}.$

Hence, the exterior derivative of $\Theta_{\lambda}$ reads,

$\displaystyle d\Theta_{\lambda}$
$\displaystyle=d\mathscr{L}\wedge\omega_{0}+d\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\wedge\omega^{\sigma}\wedge\omega_{j}+\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)d\omega^{\sigma}\wedge\omega_{j}$
$\displaystyle+d\left(\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\wedge\omega_{i}^{\sigma}\wedge\omega_{j}+\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}d\omega_{i}^{\sigma}\wedge\omega_{j}$ (4.7)
$\displaystyle=\left(\frac{\partial\mathscr{L}}{\partial
y^{\sigma}}-d_{j}\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}+d_{i}d_{j}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\omega^{\sigma}\wedge\omega_{0}-\frac{\partial}{\partial
y^{\tau}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\omega^{\sigma}\wedge\omega^{\tau}\wedge\omega_{j}$
$\displaystyle+\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{\sigma}\partial y_{kj}^{\tau}}-\frac{\partial}{\partial
y_{k}^{\tau}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\right)\omega^{\sigma}\wedge\omega_{k}^{\tau}\wedge\omega_{j}$
$\displaystyle-\frac{\partial}{\partial
y_{kl}^{\tau}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\omega^{\sigma}\wedge\omega_{kl}^{\tau}\wedge\omega_{j}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{k}^{\tau}\partial
y_{ij}^{\sigma}}\omega_{i}^{\sigma}\wedge\omega_{k}^{\tau}\wedge\omega_{j}$
$\displaystyle-\frac{\partial^{2}\mathscr{L}}{\partial y_{kl}^{\tau}\partial
y_{ij}^{\sigma}}\omega_{i}^{\sigma}\wedge\omega_{kl}^{\tau}\wedge\omega_{j}.$

From (4.5), we have

$\displaystyle dZ_{\lambda}$ $\displaystyle=d\Theta_{\lambda}$
$\displaystyle+\frac{1}{2}dP_{\sigma\nu}\wedge\omega^{\sigma}\wedge\omega^{\nu}+\frac{1}{2}P_{\sigma\nu}d\left(\omega^{\sigma}\wedge\omega^{\nu}\right)+dQ_{\sigma,\nu}^{j}\wedge\omega^{\sigma}\wedge\omega_{j}^{\nu}+Q_{\sigma,\nu}^{j}d\left(\omega^{\sigma}\wedge\omega_{j}^{\nu}\right)$
$\displaystyle+\frac{1}{2}dR_{\sigma,\nu}^{i,j}\wedge\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}+\frac{1}{2}R_{\sigma,\nu}^{i,j}d\left(\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}\right),$

and combining with (4.7) we get a decomposition of $dZ_{\lambda}$ containing
independent base terms,

$dZ_{\lambda}=E_{\lambda}+F_{0}+F_{1}+F_{2},$

where $E_{\lambda}$ is the Euler–Lagrange form of $\lambda$, $F_{0}$ and
$F_{1}$ are the $2$-contact parts and $F_{2}$ is the $3$-contact part of
$dZ_{\lambda}$. For indices $i$, $j$ running through $\\{1,2\\}$, and
$\kappa(1)=2,$ $\kappa(2)=1$, we have

$\displaystyle F_{0}$
$\displaystyle=(-1)^{j-1}\left(P_{\sigma\nu}+d_{\kappa(j)}Q_{\sigma,\nu}^{\kappa(j)}+\frac{\partial^{2}\mathscr{L}}{\partial
y^{\sigma}\partial y_{12}^{\nu}}-\frac{\partial}{\partial
y_{\kappa(j)}^{\nu}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\right)$
$\displaystyle\qquad\omega^{\sigma}\wedge\omega_{\kappa(j)}^{\nu}\wedge\omega_{j}$
$\displaystyle+(-1)^{j-1}\left(Q_{\sigma,\tau}^{\kappa(j)}+\frac{\partial}{\partial
y_{\kappa(j)\kappa(j)}^{\tau}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\right)\omega^{\sigma}\wedge\omega_{\kappa(j)\kappa(j)}^{\tau}\wedge\omega_{j}$
(4.8)
$\displaystyle+(-1)^{j-1}\left(Q_{\sigma,\tau}^{j}+2(-1)^{j}\frac{\partial}{\partial
y_{12}^{\tau}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\right)\omega^{\sigma}\wedge\omega_{12}^{\tau}\wedge\omega_{j}$
$\displaystyle+(-1)^{j-1}\left(R_{\sigma,\nu}^{\kappa(j),j}+(-1)^{j-1}\frac{\partial^{2}\mathscr{L}}{\partial
y_{\kappa(j)\kappa(j)}^{\sigma}\partial
y_{jj}^{\nu}}\right)\omega_{\kappa(j)\kappa(j)}^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{j}$
$\displaystyle+(-1)^{j-1}\left(R_{\sigma,\nu}^{j,\kappa(j)}+2(-1)^{j-1}\frac{\partial^{2}\mathscr{L}}{\partial
y_{12}^{\sigma}\partial
y_{12}^{\nu}}\right)\omega_{12}^{\sigma}\wedge\omega_{\kappa(j)}^{\nu}\wedge\omega_{j}$
$\displaystyle+(-1)^{j}R_{\sigma,\nu}^{j,j}\omega_{jj}^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{\kappa(j)}+(-1)^{j-1}R_{\sigma,\nu}^{j,j}\omega_{12}^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{j},$

and

$\displaystyle F_{1}$ $\displaystyle=\left(\frac{\partial}{\partial
y^{\sigma}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\tau}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\tau}}\right)+(-1)^{j-1}\frac{1}{2}d_{\kappa(j)}^{\prime}P_{\sigma\tau}\right)\omega^{\sigma}\wedge\omega^{\tau}\wedge\omega_{j}\quad\mathrm{Alt}(\sigma\tau)$
$\displaystyle+(-1)^{j-1}\left(d_{\kappa(j)}Q_{\sigma,\nu}^{j}+\frac{\partial^{2}\mathscr{L}}{\partial
y^{\sigma}\partial y_{jj}^{\nu}}-\frac{\partial}{\partial
y_{j}^{\nu}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\right)\omega^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{j}$
$\displaystyle-\frac{\partial}{\partial
y_{jj}^{\tau}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)\omega^{\sigma}\wedge\omega_{jj}^{\tau}\wedge\omega_{j}$
(4.9)
$\displaystyle+\frac{1}{2}(-1)^{j-1}\left(Q_{\sigma,\nu}^{\kappa(j)}-Q_{\nu,\sigma}^{\kappa(j)}+(-1)^{j}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{\kappa(j)}^{\nu}\partial
y_{12}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{\kappa(j)}^{\sigma}\partial y_{12}^{\nu}}\right)\right)$
$\displaystyle\qquad\omega_{\kappa(j)}^{\sigma}\wedge\omega_{\kappa(j)}^{\nu}\wedge\omega_{j}$
$\displaystyle+\frac{1}{2}(-1)^{j-1}\left(d_{\kappa(j)}R_{\sigma,\nu}^{j,j}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{j}^{\sigma}\partial y_{jj}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{j}^{\nu}\partial
y_{jj}^{\sigma}}\right)\omega_{j}^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{j}$
$\displaystyle+(-1)^{j-1}\left(\frac{1}{2}d_{\kappa(j)}\left(R_{\sigma,\nu}^{1,2}-R_{\nu,\sigma}^{2,1}\right)+(-1)^{j}Q_{\nu,\sigma}^{j}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{j}^{\sigma}\partial y_{12}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{\kappa(j)}^{\nu}\partial y_{jj}^{\sigma}}\right)$
$\displaystyle\qquad\omega_{1}^{\sigma}\wedge\omega_{2}^{\nu}\wedge\omega_{j},$

and

$\displaystyle F_{2}=$ $\displaystyle\frac{1}{6}\left(\frac{\partial
P_{\sigma\nu}}{\partial y^{\tau}}+\frac{\partial P_{\nu\tau}}{\partial
y^{\sigma}}+\frac{\partial P_{\tau\sigma}}{\partial
y^{\nu}}\right)\omega^{\tau}\wedge\omega^{\sigma}\wedge\omega^{\nu}$
$\displaystyle+\frac{1}{2}\left(\frac{\partial P_{\sigma\nu}}{\partial
y_{j}^{\tau}}-\frac{\partial Q_{\sigma,\tau}^{j}}{\partial
y^{\nu}}+\frac{\partial Q_{\nu,\tau}^{j}}{\partial
y^{\sigma}}\right)\omega^{\sigma}\wedge\omega^{\nu}\wedge\omega_{j}^{\tau}$
(4.10) $\displaystyle+\frac{1}{2}\frac{\partial P_{\sigma\nu}}{\partial
y_{jk}^{\tau}}\omega_{jk}^{\tau}\wedge\omega^{\sigma}\wedge\omega^{\nu}+\frac{1}{2}\frac{\partial
P_{\sigma\nu}}{\partial
y_{jkl}^{\tau}}\omega_{jkl}^{\tau}\wedge\omega^{\sigma}\wedge\omega^{\nu}$
$\displaystyle+\frac{1}{2}\left(\frac{\partial R_{\sigma,\nu}^{i,j}}{\partial
y^{\tau}}-\frac{\partial Q_{\tau,\nu}^{j}}{\partial
y_{i}^{\sigma}}+\frac{\partial Q_{\tau,\sigma}^{i}}{\partial
y_{j}^{\nu}}\right)\omega^{\tau}\wedge\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}$
$\displaystyle+\frac{\partial Q_{\sigma,\nu}^{j}}{\partial
y_{kl}^{\tau}}\omega^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{kl}^{\tau}+\frac{\partial
Q_{\sigma,\nu}^{j}}{\partial
y_{klp}^{\tau}}\omega^{\sigma}\wedge\omega_{j}^{\nu}\wedge\omega_{klp}^{\tau}$
$\displaystyle+\frac{1}{2}\frac{\partial R_{\sigma,\nu}^{i,j}}{\partial
y_{k}^{\tau}}\omega_{k}^{\tau}\wedge\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}+\frac{1}{2}\frac{\partial
R_{\sigma,\nu}^{i,j}}{\partial
y_{kl}^{\tau}}\omega_{kl}^{\tau}\wedge\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}+\frac{1}{2}\frac{\partial
R_{\sigma,\nu}^{i,j}}{\partial
y_{klp}^{\tau}}\omega_{klp}^{\tau}\wedge\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu}$

(cf. Theorem 2). As we require that $dZ_{\lambda}$ vanishes, one can now
recover concrete formulae of coefficients $P_{\sigma\nu}$,
$Q_{\sigma,\nu}^{j}$, and $R_{\sigma,\nu}^{i,j}$ in the expression (4.5) of
$Z_{\lambda}$. Indeed, annihilating (4.8) we obtain the functions (4.6), which
are correctly defined since $\mathscr{L}$ satisfies (4.2) and the symmetric
component of

$d_{\kappa(j)}Q_{\sigma,\nu}^{\kappa(j)}+\frac{\partial^{2}\mathscr{L}}{\partial
y^{\sigma}\partial y_{12}^{\nu}}-\frac{\partial}{\partial
y_{\kappa(j)}^{\nu}}\left(\frac{\partial\mathscr{L}}{\partial
y_{j}^{\sigma}}-d_{i}^{\prime}\frac{\partial\mathscr{L}}{\partial
y_{ij}^{\sigma}}\right)$

is zero.

It remains to show that all coefficients of forms $F_{1}$ (4.9) and $F_{2}$
(4.10) vanish identically. To this can be, however, directly verified by means
of the assumption on variational triviality of $\lambda$, namely conditions
(4.4) of Lemma 10, as well as partial derivative operators $\partial/\partial
y_{j}^{\tau}$, $\partial/\partial y_{jk}^{\tau}$ applied to condition (3.3).
In particular, the following identities follow from (4.4) and are employed,

$\displaystyle\frac{\partial^{3}\mathscr{L}}{\partial y_{1}^{\sigma}\partial
y_{2}^{\tau}\partial y_{12}^{\nu}}+\frac{\partial^{3}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{2}^{\sigma}\partial
y_{12}^{\tau}}+\frac{\partial^{3}\mathscr{L}}{\partial y_{1}^{\tau}\partial
y_{2}^{\nu}\partial y_{12}^{\sigma}}$
$\displaystyle\quad-\frac{\partial^{3}\mathscr{L}}{\partial
y_{1}^{\tau}\partial y_{2}^{\sigma}\partial
y_{12}^{\nu}}-\frac{\partial^{3}\mathscr{L}}{\partial y_{1}^{\sigma}\partial
y_{2}^{\nu}\partial y_{12}^{\tau}}-\frac{\partial^{3}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{2}^{\tau}\partial y_{12}^{\sigma}}=0,$
$\displaystyle\frac{\partial^{3}\mathscr{L}}{\partial y_{1}^{\nu}\partial
y_{12}^{\sigma}\partial y_{12}^{\tau}}-\frac{\partial^{3}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial y_{12}^{\nu}\partial
y_{12}^{\tau}}=0,\qquad\frac{\partial^{3}\mathscr{L}}{\partial
y_{2}^{\nu}\partial y_{12}^{\sigma}\partial
y_{12}^{\tau}}-\frac{\partial^{3}\mathscr{L}}{\partial y_{2}^{\sigma}\partial
y_{12}^{\nu}\partial y_{12}^{\tau}}=0,$
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y^{\sigma}\partial
y_{1}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial y^{\nu}\partial
y_{1}^{\sigma}}+\frac{1}{2}d_{2}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial y_{2}^{\nu}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial
y_{2}^{\sigma}}\right)+d_{1}^{\prime}d_{2}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{12}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial y_{12}^{\nu}}\right)$
$\displaystyle+\frac{1}{2}d_{2}^{\prime}d_{2}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{22}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\sigma}\partial
y_{22}^{\nu}}\right)+d_{i}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{\nu}\partial y_{1i}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y^{\sigma}\partial y_{1i}^{\nu}}\right)=0,$
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y^{\tau}\partial
y_{2}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial y^{\sigma}\partial
y_{2}^{\tau}}+\frac{1}{2}d_{1}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\tau}\partial y_{1}^{\sigma}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\sigma}\partial
y_{1}^{\tau}}\right)+\frac{1}{2}d_{1}^{\prime}d_{1}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\sigma}\partial y_{11}^{\tau}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\tau}\partial y_{11}^{\sigma}}\right)$
$\displaystyle+d_{1}^{\prime}d_{2}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\sigma}\partial y_{12}^{\tau}}-\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\tau}\partial
y_{12}^{\sigma}}\right)+d_{i}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{\sigma}\partial y_{i2}^{\tau}}-\frac{\partial^{2}\mathscr{L}}{\partial
y^{\tau}\partial y_{i2}^{\sigma}}\right)=0,$
$\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y^{\sigma}\partial
y_{12}^{\nu}}+\frac{\partial^{2}\mathscr{L}}{\partial y^{\nu}\partial
y_{12}^{\sigma}}-\frac{1}{2}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{2}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial
y_{1}^{\sigma}}\right)+2d_{1}^{\prime}d_{2}^{\prime}\frac{\partial^{2}\mathscr{L}}{\partial
y_{12}^{\nu}\partial y_{12}^{\sigma}}$ $\displaystyle-
d_{i}^{\prime}\frac{\partial^{2}\mathscr{L}}{\partial y_{i}^{\sigma}\partial
y_{12}^{\nu}}+d_{j}^{\prime}\left(\frac{\partial^{2}\mathscr{L}}{\partial
y_{2}^{\nu}\partial y_{1j}^{\sigma}}+\frac{\partial^{2}\mathscr{L}}{\partial
y_{1}^{\nu}\partial y_{2j}^{\sigma}}\right)=0.$

The equivalence of (i) and (ii) is therefore proved. ∎

######  Remark 12.

The proof of Theorem 11 shows that $Z_{\lambda}$ (4.5) satisfying the
equivalence relation “$Z_{\lambda}$ is closed if and only if $\lambda$ is
variationally trivial” does _not_ exist in general. In particular, without the
assumption (4.1) on order-reducibility of the generalized Poincaré–Cartan form
$\Theta_{\lambda}$, the coeffcients $P_{\sigma\nu}$, $Q_{\sigma,\nu}^{j}$, and
$R_{\sigma,\nu}^{i,j}$ of $Z_{\lambda}$ are generally _not_ well-defined to
ensure the closure condition of $Z_{\lambda}$ for a trivial Lagrangian
$\lambda$.

As an example illustrating this property, we refer to the Camassa–Holm
equation and its Lagrangian on the second jet prolongation of fibered manifold
$S^{1}\times\mathbb{R}\times\mathbb{R}$ over $S^{1}\times\mathbb{R}$ (see
[10]),

(4.11)
$\mathscr{L}_{CH}=\frac{1}{2}\Big{(}y_{1}(y_{2})^{2}+\frac{(y_{12})^{2}}{y_{1}}\Big{)},$

which is quadratic in second derivatives and does not satisfy the order-
reducibility condition (4.1). Following the proof of Theorem 11, it can be
directly shown that $Z_{\lambda}$ (4.5), associated to Lagrangian (4.11) and
satisfying the equivalence relation (1.1), does not exist.

Note that the result of Theorem 11 can be also directly verified as follows:
consider $Z_{\lambda}$ (4.5) with coefficients (4.6), where the associated
Lagrange function $\mathscr{L}$ is of the form $\mathscr{L}=d_{i}g^{i}$, where
(3.7) of Lemma 8 holds. One can then verify that $dZ_{\lambda}=0$ in a
straightforward way.

A possible generalization of the fundametal form, associated to a second-order
Lagrangian in field theory, has been recently studied in [19]; the obtained
result is, however, not the case described by Theorem 4 since the
corresponding Lepage form does not obey the closure condition (1.1).

######  Remark 13.

Clearly, the order-reducibility condition (4.1) imposed beforehand is
satisfied for Lagrangians $\lambda\in\Omega_{n,X}^{2}Y$ _linear in second
derivatives_ $y_{ij}^{\tau}$. In this case, if

$\mathscr{L}=A+B_{\sigma}^{ij}y_{ij}^{\sigma}=A+B_{\sigma}^{11}y_{11}^{\sigma}+2B_{\sigma}^{12}y_{12}^{\sigma}+B_{\sigma}^{22}y_{22}^{\sigma}$

is a Lagrange function satisfying (4.2), then

$\displaystyle P_{\sigma\nu}$
$\displaystyle=\frac{1}{2}\left(\frac{\partial^{2}A}{\partial
y_{1}^{\sigma}\partial y_{2}^{\nu}}-\frac{\partial^{2}A}{\partial
y_{1}^{\nu}\partial
y_{2}^{\sigma}}+\left(\frac{\partial^{2}B_{\tau}^{ij}}{\partial
y_{1}^{\sigma}\partial y_{2}^{\nu}}-\frac{\partial^{2}B_{\tau}^{ij}}{\partial
y_{1}^{\nu}\partial y_{2}^{\sigma}}\right)y_{ij}^{\tau}\right)$
$\displaystyle+(-1)^{j-1}2d_{j}^{\prime}\left(\frac{\partial
B_{\sigma}^{12}}{\partial y_{j}^{\nu}}-\frac{\partial B_{\nu}^{12}}{\partial
y_{j}^{\sigma}}\right),$ $\displaystyle Q_{\sigma,\nu}^{1}$
$\displaystyle=4\frac{\partial B_{\nu}^{12}}{\partial
y_{1}^{\sigma}}-2\frac{\partial B_{\sigma}^{12}}{\partial
y_{1}^{\nu}}-\frac{\partial B_{\sigma}^{11}}{\partial y_{2}^{\nu}},\quad
Q_{\sigma,\nu}^{2}=-4\frac{\partial B_{\nu}^{12}}{\partial
y_{2}^{\sigma}}+2\frac{\partial B_{\sigma}^{12}}{\partial
y_{2}^{\nu}}+\frac{\partial B_{\sigma}^{22}}{\partial y_{1}^{\nu}},$
$\displaystyle R_{\sigma,\nu}^{1,1}$ $\displaystyle=0,\quad
R_{\sigma,\nu}^{2,2}=0,\quad R_{\sigma,\nu}^{1,2}=0,\quad
R_{\sigma,\nu}^{2,1}=0.$

Moreover, from (3.8) it is easy to see that the order-reducibility (4.1) is a
weaker condition than the requirement on a second-order Lagrangian leading to
second-order Euler–Lagrange equations, cf. [6], which is the case of the
Einstein–Hilbert gravitation Lagrangian of General Relativity. For the example
of interaction of gravitational and electromagnetic fields Lagrangian, see
[28].

Let $\lambda\in\Omega_{2,X}^{2}Y$ be a Lagrangian for $\pi:Y\rightarrow X$
such that the generalized Poincaré-Cartan form $\Theta_{\lambda}$ (2.5) is of
second-order. We call $Z_{\lambda}$ (4.5), satisfying one of the equivalent
conditions of Theorem 11, the _fundamental Lepage equivalent_ , associated to
a second-order Lagrangian $\lambda$.

Our conjecture is that this kind of generalization of the fundamental form
using order reducibility can be provided for a general dimension of a base
manifold, which will be studied in a future work.

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