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This question was asked already on Stack Exchange under https://math.stackexchange.com/q/1114095 . It might be not on a research level, but as it could not be answered on Stack Exchange, I hope for your kind hints / references:

What do we know about the spectrum of the convolution operator in a bounded domain $D \subset \mathbb{R}$:

$$J∗u=\int_D J(x−y)u(y)dy$$

where $u$ is an $L^1(D)$ - function.

Which methods can I use to determine the spectrum?

Any hints/references are greatly appreciated.

Thank you!

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