  Based on the Sturm-Liouville eigenvalue problem, we develop a general
analytic technique to investigate the excited states of the holographic
superconductors. By including more higher order terms in the expansion of the
trial function, we observe that the analytic results agree well with the
numeric data, which indicates that the Sturm-Liouville method is very powerful
to study the holographic superconductors even if we consider the excited
states. For both the holographic s-wave and p-wave models, we find that the
excited state has a lower critical temperature than the corresponding ground
state and the difference of the dimensionless critical chemical potential
between the consecutive states is around 5. Moreover, we analytically confirm
that the holographic superconductor phase transition with the excited states
belongs to the second order, which can be used to back up the numerical
findings for both s-wave and p-wave superconductors.
