-(\[csda3\]), and related operators. Write &T\_1:=\_x\_1+\_1\_1-K\_1,\
&T\_j:=-[E]{}+\_x\_j+ \_j\_j - K\_j,j=2,3,and define a (densely defined) linear operator $T:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ by D(T):=&{L\^2(GSI)\^3 | T\_jL\^2(GSI), j=1,2,3},\
T:=&(T\_1,T\_2,T\_3). Let $f\in L^2(G\times S\times I)^3$ and $g\in T^2(\Gamma_-)^3$. The problem (\[csda1a\])-(\[csda3\]) can be expressed equivalently as the problem \[adjoint1\] T=f,\_[|\_-]{}=g,\_j(,,E\_m)=0,j=2,3.

As in section \[esols\], an application of integration by parts and the Green’s formula (\[green\]) implies \[adjoint2\] T,v\_[L\^2(GSI)\^3]{}=,T\^\*v\_[L\^2(GSI)\^3]{}vC\_0\^1(GSI\^), where $T^*v=(T_1^*v,T_2^*v,T_3^*v)$, and \[adjoint3a\] T\_1\^\*v:=&-\_x v\_1+\_1\^\*v\_1-K\_1\^\*v,\
T\_j\^\*v:=&S\_j[E]{}-\_x v\_j +\_j\^\*v\_j-K\_j\^\*v,j=2,3. Moreover, we have $\Sigma_j^*=\Sigma_j$ and for $v\in L^2(G\times S\times I)^3$, $j=1,2,3$, $$(K_j^*v)(x,\omega,E)
=\sum_{k=1}^3\int_{S\times I} \sigma_{jk}(x,\omega,\omega',E,E')\psi_k(x,\omega',E')d\omega' dE'.$$

Let $f^*\in L^2(G\times S\times I)^3$ and $g^*\in T^2(\Gamma_+)^3$. The *adjoint problem* of (\[adjoint1\]) (or equivalently (\[desol10\])-(\[desol12\])) is defined by (cf. [@agoshkov pp. 24-28]) \[adjoint4\] T\^\*\^\*=f\^\*,\_[|\_+]{}=g\^\*,\_j\^\*(,,0)=0, j=2,3, or more explicitly -\_x\_1\^\*+\_1\^\*\_1\^\*-K\_1\^\*\^\*=& f\_1\^\*, \[adesol10:1\]\
S\_j[E]{}-\_x\_j\^\* +\_j\^\*\_j\^\* - K\_j\^\*\^\*=& f\_j\^\*,j=2,3,\[adesol10:2\] holding a.e. on $G\times S\times I$, together with the *outflow* boundary and initial values $$\begin{aligned}
{3}
\psi^*_{|\Gamma_+}={}&g^* && \quad {\rm a.e.\ on}\ \Gamma_+, \label{adesol11} \\[2mm]
\psi_j^*(\cdot,\cdot,0)={}&0\ && \quad {\rm a.e.\ on}\ G\times S,\quad j=2,3. \label{adesol12}\end{aligned}$$ The adjoint problem has various kind of applications both in the existence theory of solutions and in computations. At the end of this section we give an example which is related to the dose calculation in radiation therapy. We refer also to co