On the compactness of the resolvent of a Schrödinger type singular operator with a negative parameter

Abstract

In this paper, we consider a Schrödinger-type operator with a negative parameter and a complex potential Lt=−Δ+(−t2+itb(x)+q(x)),x∈Rn,n≥1,i2=−1, where t is a parameter that arises when studying hyperbolic operators in the space L2(Rn+1). We assume with respect to the coefficients of the operator Lt that they are continuous in Rn strongly growing and rapidly oscillating functions at infinity and satisfy the condition |b(x)|≥δ0>0,q(x)≥δ>0.In the paper, under these assumptions, it is proved that there exists a bounded inverse operator for all t∈R and found a condition that ensures the compactness of the resolvent.Note that in the paper we construct regularizing operator, i.e. the question of the existence of a resolvent of an operator with singular coefficients reduces to the case of operators with smooth periodic coefficients.