  Fundamental upper bounds are given for the plasmonic multipole absorption and
scattering of a rotationally invariant dielectric sphere embedded in a lossy
surrounding medium. A specialized Mie theory is developed for this purpose and
when combined with the corresponding generalized optical theorem, an
optimization problem is obtained which is explicitly solved by straightforward
analysis. In particular, the absorption cross section is a concave quadratic
form in the related Mie (scattering) parameters and the convex scattering cross
section can be maximized by using a Lagrange multiplier constraining the
absorption to be non-negative. For the homogeneous sphere, the Weierstrass
preparation theorem is used to establish the existence and the uniqueness of
the plasmonic singularities and explicit asymptotic expressions are given for
the dipole and the quadrupole. It is shown that the optimal passive material
for multipole absorption and scattering of a small homogeneous dielectric
sphere embedded in a dispersive medium is given approximately as the complex
conjugate and the real part of the corresponding pole positions, respectively.
Numerical examples are given to illustrate the theory, including a comparison
with the plasmonic dipole and quadrupole resonances obtained in gold, silver
and aluminum nanospheres based on some specific Brendel-Bormann (BB) dielectric
models for these metals. % of gold, silver and aluminum. Based on these
BB-models, it is interesting to note that the metal spheres can be tuned to
optimal absorption at a particular size at a particular frequency.
