We introduce an efficient approach to solve Bethe equations analytically based on quantum spectral curve philosophy. This approach has two interesting highlights. First, it gives automatically only physical solutions and therefore opens a new venue for studying completeness of the Bethe Ansatz. Second, being applied to non-compact spin chains, it suggests a new combinatorial object – extended Young diagram which is capable to label all unitary representations of $U(p,q mid m)+U(1)$ algebras with integer weights. Extended Young diagrams can be bijected to ordinary Young diagrams which allows one to use ordinary Schur polynomials for analysing non-compact representations.

Trinity College Dublin