1{\leq}p{\leq}\infty$ (see\[22\],p.24).\
**Remark 4.4.** If the quasi-potential $B$ is regular , then the corresponding truncated generator $L_{\Delta}$ has a discrete spectrum.\
Further we prove that for a broad class of Levy processes the corresponding quasi-potentials $B$ are regular.\
**Example 4.1.** We consider the case when $$\phi(x)=M|x|^{-\varkappa},\quad
0<\varkappa<1.$$ **Proposition 4.1.** *Let condition $(4.11)$ be true and let the corresponding regular operator $B$ have an eigenfunction $f(x)$ with an eigenvalue $\lambda{\ne}0$. Then the function $f(x)$ is continuous.*\
*Proof.* According to Definition 4.3 there exists an integer $N(\varkappa)$ such that the kernel $\Phi_{N}(x,t)$ of the operator $$B^{N}f=\int_{\Delta}\Phi_{N}(x,y)f(y)dy,\quad f(y){\in}L^{p}(\Delta)$$ is continuous. Hence the function $f(x)$ is continuous. The proposition is proved.

The Probability of the Levy process remaining within the given domain
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In many theoretical and applied problems it is important to estimate the quantity $$p(t,\Delta)=P\{X_{\tau}{\in}\Delta\},\quad 0{\leq}\tau{\leq}t,$$ i.e. the the probability that a sample of the process $X_{\tau}$ remains inside $\Delta$ for $0{\leq}\tau{\leq}t$ (ruin problem).\
To derive the integro-differential equations corresponding to Levy processes we use the argumentation by Kac \[11\] and our own argumentation (see \[20\]-\[22\]). Now we get rid of the requirement for the process to be stable.\
Let us consider the Levy process $X_{t}$ with the continuous density (see (1.1): 