A right inverse of differential operator $\Delta+a$ in weighted Hilbert space $L^2(\mathbb{R}^n,e^{-|x|^2})$

TL;DR

This paper establishes the existence of weak solutions for a Poisson-type differential equation involving the operator  in a weighted Hilbert space with Gaussian weight, expanding understanding of such equations in weighted functional spaces.

Contribution

It provides a proof of the existence of weak solutions for a specific differential operator in a weighted Hilbert space, which was previously not well-understood.

Findings

Existence of weak solutions for  in weighted space

Extension of Poisson equation solutions to weighted Hilbert spaces

Framework for solving similar differential equations in weighted contexts

Abstract

In this note, we prove the existence of weak solutions of a Poisson type equation in the weighted Hilbert space $L^2(\mathbb{R}^n,e^{-|x|^2})$.