= Solution
Since <energy> and <angular frequency> are related by $E=\hbar\omega$, the given <eigenvalue> relation yields
$$
E_i=E_P\sqrt{\frac{1+\lambda_i}{2}}.
$$
With $E_P=4.3\ {\rm eV}$,
$$
\boxed{
E_1=4.3\sqrt{0.285}\simeq2.296\ {\rm eV},\quad
E_2=4.3\sqrt{0.325}\simeq2.451\ {\rm eV},\quad
E_3=4.3\sqrt{0.400}\simeq2.720\ {\rm eV}.}
$$
The $\lambda_i$ are dimensionless modal <eigenvalues>, not optical wavelengths despite the symbol. The same $\hbar$ multiplies both mode and bulk angular frequencies and cancels in the <energy> ratio. These are bare-core predictions of the stipulated lossless <Drude model>; adding a dielectric/metallic shell, interband response or damping changes the actual resonances.
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