  We study the curvature-induced bound states and the coherent transport
properties for a particle constrained to move on a truncated cone-like surface.
With longitudinal hard wall boundary condition, the probability densities and
spectra energy shifts are calculated, and are found to be obviously affected by
the surface curvature. The bound-state energy levels and energy differences
decrease as increasing the vertex angle or the ratio of axial length to bottom
radius of the truncated cone. In a two-dimensional (2D) GaAs substrate with
this geometric structure, an estimation of the ground-state energy shift of
ballistic transport electrons induced by the geometric potential (GP) is
addressed, which shows that the fraction of the ground-state energy shift
resulting from the surface curvature is unnegligible under some region of
geometric parameters. Furthermore, we model a truncated cone-like junction
joining two cylinders with different radii, and investigate the effect of the
GP on the transmission properties by numerically solving the open-boundary 2D
Schr\"odinger equation with GP on the junction surface. It is shown that the
oscillatory behavior of the transmission coefficient as a function of the
injection energy is more pronounced when steeper GP wells appear at the two
ends of the junction. Moreover, at specific injection energy, the transmission
coefficient is oscillating with the ratio of the cylinder radii at incoming and
outgoing sides.
