On the Bessel Operator ⊙Bt Related to the Bessel-Helmholtz and Bessel Klein-Gordon Operator

Abstract

In this paper, we study the Bessel operator ${\odot }_B^t$, iterated $t$-times and denote by $$\begin{gathered} \begin{array}{}& {\odot }_B^t={({(B_{a_1}+\cdots +B_{a_p}+m^2)}^2-{(B_{a_{p+1}}+\cdots +B_{a_{p+q}})}^2)}^t \\ \end{array} \end{gathered}$$where $p+q=n,B_{a_i}=\frac{{\partial }^2}{\partial a_i^2}+\frac{2v_i}{a_i}\frac{\partial }{\partial a_i},2v_i=2{\alpha }_i+1,{\alpha }_i>-\frac{1}{2},a_i>0$, $t\in {\mathbb{Z}}^{+}\cup \{0\}$, $m\in {\mathbb{R}}^{+}\cup \{0\}$ and $p+q=n$ is the dimension of ${\mathbb{R}}_n^{+}=\{a:a=(a_1,\dots ,a_n),a_1>0,\dots ,a_n>0\}$.