\[prop:Lc\] The linear second order operator defined by has the following features:

**(i)**

:   $L_c$ is self adjoint, i.e. $$L_c = L_c^{\dagger}$$

**(ii)**

:   $\dx\f_c$ is an eigenvector of $L_c$ with eigenvalue zero $$L_c \dx \f_c = 0$$

**(iii)**

:   $$L_c \partial_c \f_c = -\left[I - \f_c^n \dx \paren{\f_c^n\dx \cdot} \right]\paren{\f_c-1}= -c^{-1}\paren{\f_c^n -1}$$

**(iv)**

:   $$\inner{L_c \partial_c\f_c}{\partial_c \f_c}=-\frac{d}{dc}\mathcal{N}[\f_c]$$

**(v)**

:   There exists $\lambda_0<0$ and $\psi_0 \in L^2(\R)$ such that $$L_c \psi_0 = \lambda_0 \f_c^{2n} \psi_0
      \label{eq-generalized-evp}$$ i.e. $\psi_0$ is the ground state of $L_c$ with eigenvalue $\lambda_0$ of the generalized eigenvalue problem .

**(i-iv)** are trivial algebra and integration by parts. For **(v)**, note that although $L_c$ is not in Sturm-Liouville form, if, given $h$, we let $\overline{h} = \f_c^n h$, $$\f^{-n} L_c h = -c  \dx^2 \tilde{h} + \f_c^{-2n}\left[c - n\f^{n-1} -n c \f_c^{n-1}\dx\paren{\f_c^n \dx \f_c} \right] \tilde{h} = \overline{L}_c \overline{h}$$ $\overline{L}_c$ is in standard Sturm-Liouville form, and it has a zero eigenvector, $\f_c^n \dx\f_c$. Since this has one zero crossing, by oscillation theory (see [@Coddington84], amongst others), we know there exists a ground state for $\overline{L}_c$, which we will denote by $\overline{\psi_0}$ with a negative eigenvalue, $\lambda_0$. In turn, $L_c$ has a generalized eigenvector $\psi_0 = \f_c^{-n} \overline{\psi_0}$ 