Let $\;A:W^{2,2}(\mathbb R;\mathbb R^2) \to L^2(\mathbb R;\mathbb R^2)\;$ be an operator such that:
$\;Av=-v''+W_{uu}(\bar u)v\;$
where
- $\;W \in C^3(\mathbb R^2;\mathbb R)\;$ is a non-negative function with two minima at $\;a,b\;$. These minima are non-degenerate, i.e. $\;D^2 W(a):=W_{uu}(a)\;$ and $\;D^2 W(b):=W_{uu}(b)\;$ are strictly positive in the sense of quadratic forms.
- $\;\bar u:\mathbb R \to \mathbb R^2\;$ is a solution of $\;\bar u''-W_{uu}(\bar u)=0\;$ class of $\;C^4\;$ such that $\;\lim_{x \to -\infty} \bar u(x)=a\;,\;\lim_{x \to +\infty} \bar u(x)=b\;$
Then $\;Av \gt 0\;$
I was reading a paper and I came across to the above. The short explanation given is:
" by the non degeneracy of $\;W_{uu}(a)\;$ and $\;W_{uu}(b)\;$, the essential spectrum of $\;A\;$ is bounded away from $\;0\;$ and $\;ζ = \bar u'\;$ is an eigenvector of $\;A\;$ relative to the eigenvalue $\;0\;$
I'm trying to understand this explanation but I'm having a really hard time getting my head around it since I haven't seen essential spectrum before and I don't know what Theorems I might miss here.
Could somebody explain to me in more details why this operator is positive definite?
Any help would be valuable! Thanks in advance