Consider the operator $mathcal{L} : H^2(mathbb{T}_L) subset L^2(mathbb{T}_L) longrightarrow L^2(mathbb{T}_L)$ given by

$$mathcal{L} = -omega partial_x^2+3varphi^2-1,$$

that is

$$mathcal{L}(f) = -omega f”+3varphi^2f-f,$$

where $mathbb{T}_L:= mathbb{R}/Lmathbb{Z}$ and $L,w>0$ are fixeds constants and $varphi$ is a fixed function. I know that from the Floquet Theory, the operator $mathcal{}$ has only negative eigenvalue $lambda <0$, with eigenfunction $phi$.

**Question.** Why for any $psi in H^1(mathbb{T}_L)$ such that $psi perp phi$, that is $(psi, phi)_1=0$, we have

$$(mathcal{L}(phi),phi)geq 0? tag{1}$$

Here $(cdot, cdot)$ denote the inner product in $L^2$ and $(cdot, cdot)_1$ the inner product in $H^1$.