An Operator Approach to the Integration of Linear Differential Equations

TL;DR

This paper develops an operator-theoretic framework for integrating linear differential equations through intertwining relations $MT = TL$, revealing that first-order intertwiners $T = \partial + s$ reduce to Riccati-type equations for $s$, which can be linearized by the substitution $s = -({\ln h})'$. The approach extends naturally to linear partial differential equations, enabling transfer of one-dimensional results to PDEs such as the Klein–Gordon equation, by constructing transformed potentials $V(x) + 2 (\ln h)''$ and generating new solutions via $v = T u$, with Crum's formula providing a means to build $h$ from known solutions. The method yields explicit low-order intertwinings and factorization relations, clarifying the algebraic structure of intertwinings in the ring $\mathcal{F}[\partial]$ and offering a practical toolkit for producing exactly solvable operators and potentials. The work has implications for integrable systems and multi-variable extensions, presenting a transparent, constructive path for exploring Darboux-type transformations beyond ordinary differential equations.

Abstract

We develop an operator approach to the integration of linear differential equations based on intertwining relations between differential operators. Conditions for the existence of intertwining operators are obtained, and it is shown that, in low-order cases, the problem reduces to Riccati-type equations. The method is applied to linear partial differential equations, which makes it possible to construct their solutions. The linear Klein--Gordon equation is presented as an illustrative example.