Weyl's inequalities are not the full story. The characterization of the possible spectra of $A+B$, given the spectra of $A$ and $B$ is the object of A. Horn's conjecture. This is now a theorem, after hard works by Fulton, Klyatchko, Knutson, Terry Tao and others. The conjecture consists in linear inequalities (the simplest of these being Weyl's) which are found recursively on the dimension $n$. After Weyl's inequalities, there are Ky Fan or Wielandt's inequalities, etc ...

Weyl's inequalities are not the full story. The characterization of the possible spectra of $A+B$, given the spectra of $A$ and $B$ is the object of A. Horn's conjecture. This is now a theorem, after hard works by Fulton, Klyachko, Knutson, Terry Tao and others. The conjecture consists in linear inequalities (the simplest of these being Weyl's) which are found recursively on the dimension $n$. After Weyl's inequalities, there are Ky Fan or Wielandt's inequalities, etc ...