The eigenvalue problem $-{{\Delta }_{p}}u-{{\Delta }_{q}}u=\lambda {{\left| u \right|}^{q-2}}u$ with $p\,\in \,\left( 1,\,\infty \right),\,q\,\in \,\left( 2,\,\infty \right),\,p\ne \,q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ${{\mathbb{R}}^{N}}$ with $N\,\ge \,2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $\left( {{\lambda }_{1,}}+\infty \right)$ plus an isolated point $\lambda \,=\,0$. This comprehensive result is strongly related to our framework, which is complementary to the well-known case $p\,=\,q\,\ne \,2$ for which a full description of the set of eigenvalues is still unavailable.