Transcendental meromorphic solutions and the complex Schrödinger equation with delay

In this article, we focus on studying the differential-difference equation $$f^{\mathrm{\prime }}(z)=a(z)f(z+1)+R(z,f(z)),R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))},$$ where the two nonzero polynomials $P(z,f(z))$ and $Q(z,f(z))$ in $f(z)$, with small meromorphic coefficients, are coprime, and $a(z)$ is a nonzero small meromorphic function of $f(z)$. This equation includes the complex Schrodinger equation with delay as a special case. If $f(z)$ is a transcendental meromorphic solution of the equation with subnormal growth, then we derive all possible forms of the equation. Additionally, under these assumptions, we classify these specific forms based on the degrees of $P(z,f(z))$ and $Q(z,f(z))$ to establish necessary conditions for the existence of transcendental meromorphic solutions. In particular, when the degree of $P$ minus the degree of $Q$ is 2, we demonstrate that the equation reduces to a Riccati differential equation. Finally, examples are provided to support our results.