A theorem for quantum operator correspondence to the solution of the Helmholtz equation

We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V(x1,x2,x3)|x1,x2,x3x1,x2,x3|d3x = V(X1,X2,X3) = e−λ2/4:V(X1,X2,X3):, where V(x1,x2,x3) is the solution to the Helmholtz equation 2V + λ2V = 0, the symbol: : denotes normal ordering, and X1,X2,X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac's radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.