  Modern applications require a robust and theoretically solid tool for the
realistic modeling of electronic states in low dimensional nanostructures. The
$k \cdot p$ theory has fruitfully served this role for the long time since its
establishment. During the last three decades several problems have been
detected in connection with the application of the $k \cdot p$ approach to such
nanostructures. These problems are closely related to the violation of the
ellipticity conditions for the underlying model, the fact that has been largely
overlooked in the literature. We derive ellipticity conditions for $6 \times
6$, $8\times 8$ and $14 \times 14$ Hamiltonians obtained by the application of
Luttinger-Kohn theory to the bulk zinc blende (ZB) crystals, and demonstrate
that the corresponding models are non-elliptic for many common crystalline
materials. With the aim to obtain the admissible (in terms of ellipticity)
parameters, we further develop and justify a parameter rescaling procedure for
$8\times 8$ Hamiltonians. This allows us to calculate the admissible parameter
sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, InN. The
newly obtained parameters are then optimized in terms of the bandstructure fit
by changing the value of the inversion asymmetry parameter $B$ that is proved
to be essential for ellipticity of $8\times 8$ Hamiltonian. The consecutive
analysis, performed here for all mentioned $k \cdot p$ Hamiltonians, indicates
the connection between the lack of ellipticity and perturbative terms
describing the influence of out-of-basis bands on the structure of the
Hamiltonian. This enables us to quantify the limits of models' applicability
material-wise and to suggest a possible unification of two different $14 \times
14$ models, analysed in this work.
