On the Green function of the (⊕+m²)^k operator

Abstract

In this article, we study the Green function of the operator (⊕+m ²) ^k which is iterated k-times and is defined by

where m is a positive real number and p+q=n is the dimension of the n-dimensional Euclidean space ℝ ⁿ , x=(x ₁, x ₂, …, x _n )∈ℝ ⁿ and k is a nonnegative integer. At first, we study the elementary solution or Green function of the operator (⊕+m ²) ^k . Moreover, the operator (⊕+m ²) ^k can be related to the ultra-hyperbolic Klein–Gordon operator (□+m ²) ^k , the Helmholtz operator (▵+m ²) ^k and the diamond operator of the form (♦+m ²) ^k , and also we obtain the elementary solutions of such operators. We also apply such a Green function to obtain the solution of the equation (⊕+m ²) ^k U(x)=f(x), where f is a generalized function and U(x) is an unknown function for x∈ℝ ⁿ .

Keywords:

• Ultra-hyperbolic Klein–Gordon operator
• Elementary solution
• Helmholtz operator
• Diamond operator

Acknowledgements

The authors would like to thank The Thailand Research Fund and Graduate School, Chiang Mai University, Thailand for financial support. Jessada Tariboon is supported by The Royal Golden Jubilee Project Grant No. PHD/0072/2547.