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, x3 x1, x2, x3| d3 x = 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.     

Wavelet transform for Fresnel-transformed mother wavelets

In this paper, we propose the so-called continuous Fresnel-wavelet combinatorial transform which means that the mother wavelet undergoes the Fresnel transformation. This motivation can let the mother-wavelet-state itself vary from |ψ> to F_r,s⁺ |ψ> , except for variation within the family of dilations and translations. The Parseval's equality, admissibility condition and inverse transform of this continuous Fresnel-wavelet combinatorial transform are analysed. By taking certain parameters and using the admissibility condition of this continuous Fresnel-wavelet combinatorial transform, we obtain some mother wavelets. A comparison between the newly found mother wavelets is presented.