Consider the following integral operator:
$$K(f) : x \mapsto\int_0^1 K(x,x')f(x') dx', \quad \text{where} \quad K(x,x') = - |x-x'|^{3/2}.$$
The kernel is sometimes referred to as a negative distance kernel in machine learning (since it is given as a power of the distance). My setting is one-dimensional.
I am interested in proving that the asymptotic behavior of the spectrum is $\lambda_n \sim c \cdot n^{-5/2}$, where $c > 0$ is some numerical constant. Note that this statement includes two facts (asymptotic eigenvalues are positive and decay at a given pace). These facts are consistent both with my intuition of the operator and with numerical computations I have ran.
Does someone know about references tackling similar questions?
Thanks!
