  We consider a Josephson junction built with the two-dimensional semi-Dirac
semimetal, which features a hybrid of linear and quadratic dispersion around a
nodal point. We model the weak link between the two superconducting regions by
a Dirac delta potential because it mimics the thin-barrier-limit of a
superconductor-barrier-superconductor configuration. Assuming a homogeneous
pairing in each region, we set up the BdG formalism for electronlike and
holelike quasiparticles propagating along the quadratic-in-momentum dispersion
direction. This allows us to compute the discrete bound-state energy spectrum
$\varepsilon $ of the subgap Andreev states localized at the junction. In
contrast with the Josephson effect investigated for propagation along linearly
dispersing directions, we find a pair of doubly degenerate Andreev bound
states. Using the dependence of $\varepsilon $ on the superconducting phase
difference $\phi$, we compute the variation of Josephson current as a function
of $\phi$.
