Leaving principle for Bloch eigenvalues of an optical Schrödinger operator

Let $H(c)$ be the operator with pure imaginary potential $2ic\cos 2x$, and let $L(V)$ be the corresponding operator whose Bloch eigenvalues depend on $V$. Leaving principle. If $c$ increases from $0$ to $\infty$, then all Bloch eigenvalues of $H(c)$ leave the real line. Equivalently, if $V$ increases from $1/2$ to $\infty$, then all Bloch eigenvalues of $L(V)$ leave the real line. The principle was proved for the Bloch eigenvalues $\lambda_1(t,V)$ and $\lambda_2(t,V)$, but the general assertion is resolved according to the source.