Spectrum of a self-adjoint operator in $L_2(K)$, where $K$ is a local field; analog of the Feynman–Kac formula

We consider operators in $L_2(K)$, where $K$ is a local field that is a sum of the operator of convolution with a generalized function and multiplication by a function. A criterion of self-adjointness is given, and some results on the discrete spectrum are obtained. An analog of the Feynman–Kac formula is derived.