# fa.functional analysis – Property \$(mathcal{L}(phi),phi)geq 0\$ about a operator \$mathcal{L}\$

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 $$H^1$$.