Linear differential equations with solutions in weighted Fock spaces

TL;DR

This paper investigates the conditions under which solutions to linear complex differential equations belong to weighted Fock spaces, focusing on the relationship between coefficients and solutions, especially for second-order equations.

Contribution

It provides new sufficient conditions using Bergman kernels for solutions of differential equations to be in weighted Fock spaces, extending understanding of their functional properties.

Findings

Derived conditions for solutions in weighted Fock spaces

Analyzed relations between coefficients and solutions

Focused on second-order differential equations

Abstract

This research is concerned with the nonhomogeneous linear complex differential equation $$ f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_{1}f'+A_{0}f=A_{k} $$ in the complex plane. In the higher order case, the mutual relations between coefficients and solutions in weighted Fock spaces are discussed, respectively. In particular, sufficient conditions for the solutions of the second order case $$ f"+Af=0 $$ to be in some weighted Fock space are given by Bergman reproducing kernel and coefficient $A$.