Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories

We study the semiclassical distribution of resonances of a $2\times 2$ matrix Schrödinger operator (1.1) near a fixed energy level where the underlying classical trajectories ${\mathrm{\Gamma }}_1$ of $P_1$ and ${\mathrm{\Gamma }}_2$ of $P_2$ are, respectively, periodic and non-trapping. The aim is to compute the imaginary part of the resonances appearing near the eigenvalues created by $P_1$ when ${\mathrm{\Gamma }}_1$ intersects ${\mathrm{\Gamma }}_2$ with finite contact order $m$. Recent results [M. Assal, S. Fujiié and K. Higuchi, Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories, Int. Math. Res. Not.2024 (2024) 6879–6905] and [M. Assal, S. Fujiié and K. Higuchi, Transition of the semiclassical resonance widths across a tangential crossing energy-level, J. Math. Pures Appl.191 (2024) 103634] assert that the width of resonances is of polynomial order in the semiclassical parameter with exponent $1+2/(m+1)$, if the interaction $U=r_0(x)+ih{r}_1(x)D_x$ is elliptic at the crossing point. Here, we remove this ellipticity assumption and prove that the exponent is $1+2(k+1)/(m+1)$, where $k$ is a “vanishing order” of the interaction at the crossing points, suitably defined in terms of the vanishing orders of $r_0$ and $r_1$ depending on whether the crossing point is a turning point or not.