Quark-anti-quark potential in N $$ mathcal{N} $$ = 4 SYM

We construct a closed system of equations describing the quark-anti-quark potential at any coupling in planar $$ mathcal{n} $$ = 4 supersymmetric yang-mills theory. it is based on the quantum spectral curve method supplemented with a novel type of asymptotics. we present a high precision numerical solution reproducing the classical and one-loop string predictions very accurately. we also analytically compute the first 7 nontrivial orders of the weak coupling expansion. moreover, we study analytically the generalized quark-anti-quark potential in the limit of large imaginary twist to all orders in perturbation theory. we demonstrate how the qsc reduces in this case to a one-dimensional schrodinger equation. in the process we establish a link between the q-functions and the solution of the bethe-salpeter equation.