Asymptotic form of the spectrum of operators associated with $p$-adic fields

We consider a locally compact nonconnected nondiscrete field and study a linear operator given by the sum of the operator of multiplication by a function and the operator of convolution with a generalized function. We derive the asymptotic form of the spectrum of that linear operator. In this problem, we use the generalized $p$-adic Feynman–Kac formula.