We give a derivation of quantum spectral curve (QSC) - a finite set of
Riemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposed
in our recent paper Phys.Rev.Lett. 112 (2014). We also generalize this
construction to all local single trace operators of the theory, in contrast to
the TBA-like approaches worked out only for a limited class of states. We
reveal a rich algebraic and analytic structure of the QSC in terms of a so
called Q-system -- a finite set of Baxter-like Q-functions. This new point of
view on the finite size spectral problem is shown to be completely compatible,
though in a far from trivial way, with already known exact equations (analytic
Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to
demonstrate how the classical finite gap solutions and the asymptotic Bethe
ansatz emerge from our formalism in appropriate limits.Comment: 96 pages, 15 figures; Some mathematica examples added in v2;
Published version is v
