I am physics PhD student working on quantisation of electromagnetic fields in a non-homogeneous media. I am working through a paper at the moment and I am struggling with one of the statements. In the paper, there is a time independent eigenvalue problem obtained from inserting a sum of exponential functions of time into the Maxwell equations.
$$\pmb{\nabla}\times\left(\frac{\pmb{\nabla}\times\pmb{B}_m(\pmb{r})}{n^2(\pmb{r})}\right) = \frac{\omega_m^2}{c^2}\pmb{B}_m(\pmb{r})$$
The text calls this a "second order Hermitian eigenvalue problem". I am unsure how to demonstrate that this problem is Hermitian and would like some help, I will attach the paper below for clarity but only the relevant couple of pages as it was sent to me by my supervisor and I am don't think it is appropriate to distribute the whole thing. I first tried establishing an inner product integrating over a large box of volume $V$ with periodic boundary conditions.
$$\hat{O}(\pmb{Y}) = \pmb{\nabla}\times\left(\frac{\pmb{\nabla}\times\pmb{Y}(\pmb{r})}{n^2(\pmb{r})}\right) \\ (\pmb{X},\hat{O}\pmb{Y}) = \iiint_{V}d^3\pmb{r} \pmb{X}\cdot\hat{O}(\pmb{Y}) = \iiint_{V}d^3\pmb{r} \pmb{X}\cdot\pmb{\nabla}\times\left(\frac{\pmb{\nabla}\times\pmb{Y}(\pmb{r})}{n^2(\pmb{r})}\right)$$
I then tried to find the components of the operator $\hat{O}$ acting on $\pmb{Y}$ to determine the inner product. I have used implicit summation over repeated symbols.
$$\left[\pmb{\nabla}\times\left(\frac{\pmb{\nabla}\times\pmb{Y}(\pmb{r})}{n^2(\pmb{r})}\right)\right]_k = \epsilon_{ijk}\frac{\partial}{\partial x_i}\left(\frac{1}{n^2(\pmb{r})}\left[\pmb{\nabla}\times\pmb{Y}(\pmb{r})\right]_j\right) = \epsilon_{ijk}\frac{\partial}{\partial x_i}\left(\frac{1}{n^2(\pmb{r})}\epsilon_{mnj}\frac{\partial}{\partial x_m} Y_n(\pmb{r})\right) \\ = -\epsilon_{jik}\epsilon_{jmn}\left(-\frac{2}{n^3(\pmb{r})}\frac{\partial n(\pmb{r})}{\partial x_i}\frac{\partial Y_n(\pmb{r})}{\partial x_m} + \frac{1}{n^2(\pmb{r})}\frac{\partial^2 Y_n(\pmb{r})}{\partial x_i\partial x_m} \right) \\ = -\left(\delta_{im}\delta_{kn} - \delta_{in}\delta_{km}\right)\left(-\frac{2}{n^3(\pmb{r})}\frac{\partial n(\pmb{r})}{\partial x_i}\frac{\partial Y_n(\pmb{r})}{\partial x_m} + \frac{1}{n^2(\pmb{r})}\frac{\partial^2 Y_n(\pmb{r})}{\partial x_i\partial x_m} \right) \\ = \frac{2}{n^3(\pmb{r})}\frac{\partial n(\pmb{r})}{\partial x_m}\frac{\partial Y_k(\pmb{r})}{\partial x_m} - \frac{1}{n^2(\pmb{r})}\frac{\partial^2 Y_k(\pmb{r})}{\partial x_m\partial x_m} - \frac{2}{n^3(\pmb{r})}\frac{\partial n(\pmb{r})}{\partial x_n}\frac{\partial Y_n(\pmb{r})}{\partial x_k} + \frac{1}{n^2(\pmb{r})}\frac{\partial}{\partial x_k}\frac{\partial Y_n(\pmb{r})}{\partial x_n}$$
From here I tried to integrate the inner product of another vector function with this vector function.
$$(\pmb{X},\hat{O}\pmb{Y}) = \iiint_{V}d^3\pmb{r} \pmb{X}\cdot\pmb{\nabla}\times\left(\frac{\pmb{\nabla}\times\pmb{Y}(\pmb{r})}{n^2(\pmb{r})}\right) \\ = \iiint_{V}d^3\pmb{r} X_k\left(\frac{2}{n^3(\pmb{r})}\frac{\partial n(\pmb{r})}{\partial x_m}\frac{\partial Y_k(\pmb{r})}{\partial x_m} - \frac{1}{n^2(\pmb{r})}\frac{\partial^2 Y_k(\pmb{r})}{\partial x_m\partial x_m} - \frac{2}{n^3(\pmb{r})}\frac{\partial n(\pmb{r})}{\partial x_n}\frac{\partial Y_n(\pmb{r})}{\partial x_k} + \frac{1}{n^2(\pmb{r})}\frac{\partial}{\partial x_k}\frac{\partial Y_n(\pmb{r})}{\partial x_n}\right)$$
I want to demonstrate:
$$(\pmb{X},\hat{O}\pmb{Y}) = (\pmb{Y},\hat{O}^*\pmb{X})$$ So it tried to do this by parts but it won't work due to the reciprocal of the refractive index, $n$. Are there any other ways I can prove this, any help would be appreciated. Thank you.