that $$g(a_{k})=g(b_{k})=g^{\prime}(a_{k})=g^{\prime}(b_{k})=0,\quad 1{\leq}k{\leq}n,
\quad g^{\prime\prime}(x){\in}L^{p}(\Delta),\quad p>1.$$We introduce the operator $P_{\Delta}$ by relation $P_{\Delta}f(x)=f(x)$ if $x{\in}\Delta$ and $P_{\Delta}f(x)=0$ if $x{\notin}\Delta.$\
**Definition 4.1.** The operator $$L_{\Delta}=P_{\Delta}LP_{\Delta}$$ is called a *truncated generator*.\
**Definition 4.2.** The operator $B$ with the definition domain dense in $L^{p}(\Delta)$ is called *a quasi-potential* if the functions $f=Bg$ belong to definition domain of $L_{\Delta}$ and $$-L_{\Delta}f=g.$$ It follows from (4.3) that $$-P_{\Delta}Lf=g,\quad ( f=Bg).$$ **Remark 4.1.** In a number of cases (see the next section) we need relation (4.4). In these cases we can use the quasi-potential $B$, which is often simpler than the corresponding potential $Q$.\
**Remark 4.2.** The operators of type (4.2) are investigated in book (\[22\],Ch.2).\
From relation (4.3) we deduce that $$Bg{\ne}0,\quad if\quad g{\ne}0,\quad
g{\in}L^{p}(\Delta).$$ **Definition 4.3.** We call the operator $B$ a *regular* if the following conditions are fulfilled.\
1). The operator $B$ is compact and has the form $$Bf=\int_{\Delta}\Phi(x,y)f(y)dy,\quad f(y){\in}L^{p}(\Delta), \quad
p{\geq}1,$$ where the function $\Phi(x,y)$ can have a discontinuity only when $x=y.$\
2). There exists a function $\phi(x)$ such that $$|\Phi(x,y)|{\leq}\phi(x-y),$$ $$\int_{-R}^{R}\phi(x)dx<\infty \quad if \quad
0<R<\infty.$$ 3).$$\Phi(x,y){\geq}0,\quad
x,y{\in}\Delta,$$ $$\Phi(a_{k},y)=\Phi(b_{k},y)=0,\quad 1{\leq}k{\leq}n.$$ 4). Relation (4.5) is valid.\
**Remark 4.3.** In view of condition (4.7) the regular operator $B$ is bounded in the spaces $L^{p}(\Delta),\quad 