  The double-well potential is a good example, where we can compute the
splitting in the bound state energy of the system due to the tunneling effect
with various methods, namely WKB or instanton calculations. All these methods
are non-perturbative and there is a common belief that it is difficult to find
the splitting in the energy due to the barrier penetration from a perturbative
analysis. However, we will illustrate by explicit examples containing singular
potentials (e.g., Dirac delta potentials supported by points and curves and
their relativistic extensions)that it is possible to find the splitting in the
bound state energies by developing some kind of perturbation method.
