Lie symmetries of fundamental solutions to the Leutwiler-Weinstein equation (Q6170117)

A constructive method to find fundamental solutions to equations of the form \(\sum_{|\alpha|\leq m}a_\alpha (x)\, D^\alpha u\!=\!0\), applicable when the equation has enough symmetry, previously introduced by the first author, is presented in detail in the first part of the article. A fundamental solution of the Leutwiler-Weinstein equation \(Lu:=\Delta u\!+\!\frac{k}{x^n}\frac{\partial u}{\partial x^n}\!+\!\frac {\ell}{(x^n)^2}u\!=\!0\) in \(\mathbb{R}_+^n\), where \(k\), \(\ell\) are two real parameters, is explicitely obtained by using this method. By starting from the infinitesimal generators of \(Lu\!=\!0\), the authors determine the symmetries of \(Lu\!=\!\delta(x\!-\!x_0)\) and use the invariant solutions. The possibility to use the obtained fundamental solution to find the Green's function is investigated