  Deviations from the uniform oscillator spacing, related to the shape of the
confining potential, have a strong influence on few-electron states in quantum
dots when Coulomb effects are included. Distinct signatures are found for level
spacings increasing (pot shape) and decreasing (cone shape) with energy. Cone
deformations affect the levels near the ground state, such that observable
effects are predicted. Pot deformations partially negate the effect of the
Coulomb force, thus their spectra are similar to those of a perfect oscillator
with a smaller Coulomb repulsion. The Coulomb force is treated by exact
diagonalization, for which purpose efficient closed-form expressions for the
matrix elements are derived. The Coulomb matrix element in relative coordinates
is reduced to a single sum over four binomial coefficients, for which simple
analytic approximations are found.

Mesoscale and Nanoscale Physics