Lagrangian Schur index and Bethe ansatz type formula

We propose a surprisingly elementary method to compute the Schur index in closed-form for general $\mathcal{N}=2$ Lagrangian theories. The method is inspired by the Bethe ansatz type formula for $\mathcal{N}=1$ superconformal index. We identify issues underlying the original derivation: the loss of periodicity property upon integration and the omitted poles outside of the annulus region. We circumvent the problems and transform integration into solving a simple difference equation. The final result is expressed as a finite sum of quasi-Jacobi forms involving twisted Eisenstein series. We test our method on different types of theories, including BCD-type $\mathcal{N}=4$ theories and various $\mathcal{N}=2$ quiver gauge theories.