Regge trajectories from the adjoint sector of Matrix Quantum Mechanics

We reexamine the large $N$ limit of SU$(N)$ symmetric quantum mechanics of a Hermitian matrix whose singlet sector is well known to be exactly solvable via free fermions. When the Fermi level approaches a maximum of the potential, there is critical behavior corresponding to string theory in two dimensions. We uncover new phenomena in the adjoint sector by solving the Marchesini-Onofri equation both numerically and analytically using semiclassical approximations: at criticality, the spectrum is governed by Regge trajectories with energy eigenvalues growing according to ${\mathrm{\Delta }}^2\sim n\mathrm{/}{\alpha }^{\mathrm{\prime }}$. In the dual 2D string theory, we interpret these states as oscillatory excitations of a short'' folded open string. Up to sub-leading corrections, this Regge behavior is essentially universal and is insensitive to the particular potential we choose to approach criticality. Slightly away from criticality, the highly excited states transition into long strings’’ that extend far into the Liouville direction.