I have the follwoing integral operator acting on $L^{2}(\mathbb{R})$. $$L_{k}: \phi(x) \rightarrow \int \frac{a}{\pi}e^{-\alpha(x^2+y^2)}e^{-t\beta(x-y)^2} \phi(y)dy$$
I would like to compute the operator norm.
$$ \|L_{K}\|:= sup _{\substack{ \phi \in L^{2}(\mathbb{R}), \\ \|\phi\|_{L^{2}(\mathbb{R})} = 1}}\|L_{K}\phi\|_{L^{2}(\mathbb{R})}$$
$L_{K}$ is self adjoint so the operator norm is equal to the spectral radius.
$$ \|L_{K}\| = max_{n}\{|\lambda_{n}|\}$$ where we maximize over the spectrum of $L_{K}$.
I have tried to decompose the kernel $K(x,y) := \frac{a}{\pi}e^{-\alpha(x^2+y^2)}e^{-t\beta(x-y)^2}$ using Hermite fucntions but this has been fruitless. Owing to the simplicity of the kernel I was hoping that there would be a symple way of getting an exact value for the operator norm. I would appreciate any comments and or help with computing this operator norm. Thank you very much for your time.