  This work introduces the families of generalized energy operators
$([[.]^p]_k^+)_{k\in\mathbb{Z}}$ and $([[.]^p]_k^-)_{k\in\mathbb{Z}}$ ($p$ in
$\mathbb{Z}^+$). One shows that with Lemma 1, the successive derivatives of
$\big ([[$f$]^{p-1}]_1^+ \big)^n$ ($n$ in $\mathbb{Z}$, $n\neq 0$) can be
decomposed with the generalized energy operators $\big
([[.]^p]_k^+\big)_{k\in\mathbb{Z}}$ when $f$ is in the subspace
$\mathbf{S}_p^-(\mathbb{R})$. With Theorem 1 and $f$ in
$\mathbf{s}_p^-(\mathbb{R})$, one can decompose uniquely the successive
derivatives of $\big ([[$f$]^{p-1}]_1^+ \big)^n$ ($n$ in $\mathbb{Z}$, $n\neq
0$) with the generalized energy operators $\big
([[.]^p]_k^+\big)_{k\in\mathbb{Z}}$ and $\big
([[.]^p]_k^-\big)_{k\in\mathbb{Z}}$. $\mathbf{S}_p^-(\mathbb{R})$ and
$\mathbf{s}_p^-(\mathbb{R})$ ($p$ in $\mathbb{Z}^+$) are subspaces of the
Schwartz space $\mathbf{S}^-(\mathbb{R})$. These results generalize the work of
[arxiv:1208.3385]. The second fold of this work is the application of the
generalized energy operator families onto the solutions of linear partial
differential equations. The solutions are functions of two variables and
defined in subspaces of $\mathbf{S}^-(\mathbb{R}^2)$. The theory is then
applied to the Helmholtz equation. In this specific case, the use of
generalized energy operators in the general solution of this PDE extends the
results of [montilletIMF45-48-2010]. This work ends with some numerical
examples. We also underline that this theory could possibly open some
applications in astrophysics and aeronautics.
