Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator

The distribution of poles of zero-parameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics $-\sqrt{z/6}+O(1)$ as $z \to \infty$, $|\arg z|<4\pi/5$. At the sector $|\arg z|>4\pi/5$ it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for $|z|<\text{const}$ allowed B. Dubrovin to make a conjecture that all poles of the intégrale tritronquée belong to this sector. As a step to prove this conjecture, we study the Riemann–Hilbert (RH) problem related to the specified solution of the Painlevé I equation. It is "undressed" to a similar RH problem for the Schrödinger equation with cubic potential. The latter determines all coordinates of poles for the intégrale tritronquée via a Bohr–Sommerfeld quantization conditions.