# eigenvalues and eigenfunctions for a nondegenerate $L^2$ kernel

Consider the symmetric $$L^2$$ kernel $$K(x,t)=\log(1-\cos(x-t))$$ for $$0\leq x,t\leq 2\pi$$ . Find the eigenvalues and corresponding eigenfunctions of $$K(x,t)$$

By general procedure we consider a Fredholm equation $$y(x)=\lambda\int_0^{2\pi}\log(1-\cos(x-t))y(t)dt$$ $$\implies y(x)=\lambda\log 2\int_0^{2\pi}y(t)dt+2\lambda\int_0^{2\pi}\log\sin\left(\frac{x-t}{2}\right)y(t)dt$$ $$\implies y'(x)=2\lambda\int_0^{2\pi}\cot\left(\frac{x-t}{2}\right)y(t)dt$$ But in this way I'm unable to convert the equation into a Sturm-Liouville boundary value problem and then find the eigenvalues and eigenfunctions . What is the alternative way to find the same for this problem ? Any help is appreciated .

• An angle of attack could be via the Fourier series $\ln(\sin(x))=\ln(2)-\sum_{n=1}^{\infty} \frac1n \cos(2nx)$ Nov 18, 2021 at 12:28
• Does this answer your question? $K(x,t)=-2 \log \left( \sin \left( \frac{1}{2} \left( x-t \right) \right) \right)$ eigenvalues and eigenfunctions Nov 18, 2021 at 12:40
• See as well math.stackexchange.com/q/3963548 in order to show that the transformation made by the OP of the other question is the same as yours. (references found with the formula searching tool approach0.xyz) Nov 18, 2021 at 12:41
• Any comment ? Don't forget that this site is based on dialog... Nov 19, 2021 at 18:57