On the Bessel Operator $\odot_{B}^{t}$ 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 $$\odot_{B}^{t}= \left(\left(B_{a_{1}}+\cdots+B_{a_{p}}+m^{2}\right)^{2} - \left(B_{a_{p+1}}+\cdots+B_{a_{p+q}}\right)^{2}\right)^{t} \nonumber\\
$$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 $\mathbb{R}_{n}^{+}=\{ a:a=(a_{1},\ldots, a_{n}), a_{1}>0,\ldots, a_{n}>0\}$.