  We analyze the superconvergence properties of ultra-weak discontinuous
Galerkin (UWDG) methods with various choices of flux parameters for
one-dimensional linear Schr\"odinger equation. In our previous work [10],
stability and optimal convergence rate are established for a large class of
flux parameters. Depending on the flux choices and if the polynomial degree $k$
is even or odd, in this paper, we prove $2k$ or $(2k-1)$-th order
superconvergence rate for cell averages and numerical flux of the function, as
well as $(2k-1)$ or $(2k-2)$-th order for numerical flux of the derivative. In
addition, we prove superconvergence of $(k+2)$ or $(k+3)$-th order of the DG
solution towards a special projection. At a class of special points, the
function values and the first and second order derivatives of the DG solution
are superconvergent with order $k+2, k+1, k$, respectively. The proof relies on
the correction function techniques initiated in [8], and applied to [6] for
direct DG (DDG) methods for diffusion problems. Compared with [6],
Schr\"odinger equation poses unique challenges for superconvergence proof
because of the lack of the dissipation mechanism from the equation. One major
highlight of our proof is that we introduce specially chosen test functions in
the error equation and show the superconvergence of the second derivative and
jump across the cell interfaces of the difference between numerical solution
and projected exact solution. This technique was originally proposed in [12]
and is essential to elevate the convergence order for our analysis. Finally, by
negative norm estimates, we apply the post-processing technique and show that
the accuracy of our scheme can be enhanced to order $2k.$ Theoretical results
are verified by numerical experiments.
