  A detailed analytical inspection of light scattering by a particle with high
refractive index m+i\kappa and small dissipative constant \kappa is presented.
We have shown that there is a dramatic difference in the behavior of the
electromagnetic field within the particle (inner problem) and the scattered
field outside it (outer problem). With an increase in m at fix values of the
other parameters, the field within the particle asymptotically converges to a
periodic function of m. The electric and magnetic type Mie resonances of
different orders overlap substantially. It may lead to a giant concentration of
the electromagnetic energy within the particle. At the same time, we
demonstrate that identical transformations of the solution for the outer
problem allow to present each partial scattered wave as a sum of two
partitions. One of them corresponds to the m-independent wave, scattered by a
perfectly reflecting particle and plays the role of a background, while the
other is associated with the excitation of a sharply-m-dependent resonant Mie
mode. The interference of the partitions brings about a typical asymmetric Fano
profile. The explicit expressions for the parameters of the Fano profile have
been obtained "from the first principles" without any additional assumptions
and/or fitting. In contrast to the inner problem, at an increase in m the
resonant modes of the outer problem die out, and the scattered field converges
to the universal, m-independent profile of the perfectly reflecting sphere.
Numerical estimates of the discussed effects for a gallium phosphide particle
are presented.
