For simplicity, we have taken $H$ to be real. Let's now deform this Hamiltonian into another Hamiltonian $H'$, also real. We can imagine that this deformation describes the changes that occur to the dot as an external parameter, such as a gate voltage, is varied. We can parameterize the deformation by

$H(\alpha) = \alpha H' + (1-\alpha) H,$

$$

H(\alpha) = \alpha H' + (1-\alpha) H,

$$

so that at $\alpha=0$ we are at the initial Hamiltonian and at $\alpha=1$ we are at the final Hamiltonian. Let's see what the energy levels do as a function of $\alpha$ (we use more levels here than in the matrix above so that the spectrum looks more interesting).