Poles of intégrale tritronquée and anharmonic oscillators. Asymptotic localization from WKB analysis

Poles of intégrale tritronquée are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We show that the poles are well approximated by solutions of a pair of Bohr–Sommerfeld quantization conditions (the Bohr–Sommerfeld–Boutroux (B–S–B) system): the distance between a pole and the corresponding solution of the B–S–B system vanishes asymptotically