Local and Non-Local Potentials

Can anybody explain the difference and concepts between local and non-local potentials in light of quantum mechanics?

• If you write a potential like $V(x)$ applied to your wave functions $\psi(x)$, that is local. Non-local could be something like $V(x+5cm)$ or some integral term. – Luke Apr 26 '18 at 12:54
• In what context have you encountered these terms? – ACuriousMind Apr 26 '18 at 16:42
• @ACuriousMind: My question refers to the remark at page 265 in Schwabl's "Advanced Quantum Mechanics": the theory is local if $\mathcal{L}(x)$ if $\phi_r(x)$ depends only at $x$. What is an example for a non local potential? – KarlPeter Apr 27 '18 at 8:58
• @Luke: By integral term you mean something like $\int f(x, x') dx'$? But this can also be written at $V(x)$, or not, since the extra variable $x'$ is integrated. Or do you mean the expression integral term in another way? – KarlPeter Apr 27 '18 at 9:02
• @KarlPeter: I mean a term like in the following equation: $i \partial_t \psi(t,x) = H^{free} \psi(t,x) + \int_{\mathbb{R}} dx' V(x,x')\psi(t,x')$. – Luke Apr 27 '18 at 12:33