An estimate for the resolvent of a non-selfadjoint differential operator on an unbounded domain

We consider the operator T defined by (Tf)(x) = (Sf)(x)+q(x)f (x), x € ω, where ω ∪ Rⁿ is an unbounded domain, S is a positive definite selfadjoint operator defined on a domain Dom(S) ∪ L ²(ω)and q(x)is a bounded complex measurable function with the property Im q(x)€ L^v(ω)for v € (1,∞). We derive an estimate for the norm of the resolvent of T. In addition, we prove that T is invertible, and the inverse operator T^-1 is a sum of a normal operator and a quasinilpotent one, having the same invariant subspaces. By the derived estimate, spectrum perturbations are investigated. Moreover, a representation for the resolvent of T by the multiplicative integral is established. As examples, we consider the Schrödinger operators on the positive half-line and orthant.