  Very strong electromagnetic field can be generated in peripheral relativistic
heavy ion collisions. This work is devoted to exploring the interplay between
the effects of a constant external electric field and confining potential on
heavy-flavor mesons. As the corresponding vector potential linearly depends on
one spatial coordinate for a constant electric field, it might be able to
overcome the linear confining potential of QCD and induce deconfinement. To
perform analytic calculations and for comparison, one and two dimensional
systems are studied together with the realistic three dimensional systems. The
one dimensional Schr$\ddot{\text o}$dinger equation can be solved analytically
with the help of Airy functions, and deconfinement is indeed realized when the
electric field is larger than the string tension. Focus on the confining case,
the two and three dimensional Schr$\ddot{\text o}$dinger equations can be
solved analytically in large $r$ limit with the help of elliptic cosine/sine
functions, and the wave functions are dominated by the region antiparallel to
the electric field. When a more realistic potential is applied, a non-monotonic
feature is found for $\Upsilon(2S)$ and $\Upsilon(3S)$-like mesons with
increasing electric field.
