Here are two excerpts
Schrödinger's equation $$iu_t+\Delta u=0$$ Lawrence C. Evans, Partial Differential Equations (2010)
and
Physically, the initial-value problem for the linear Schrödinger equation $$u_t=i\Delta u+\phi,\qquad t\in\Bbb R,\quad x\in\Bbb R^3,\quad i=\sqrt{-1},\quad u(0,x)=f(x)$$ [...]
Weihrauch and Zhong, Computing Schrödinger propagators on Type-2 Turing machines (2006)
Notably, neither of these two forms have the potential term, unless I'm missing something. Is it standard to talk about the "Schrödinger equation" as not having a potential term? In the second image, there is a so-called "forcing term", which I've never seen in physical settings, but there isn't a potential term, which is almost always present in physically relevant problems. What's going on there?