Perturbation determinants for singular perturbations

Let A be a densely defined symmetric operator and let {Ã′, Ã} be an ordered pair of proper extensions of A such that their resolvent difference is of trace class. We study the perturbation determinant Δ_Ã′/Ã(·) of the singular pair {Ã′, Ã} by using the boundary triplet approach. We show that, under additional mild assumptions on {Ã′, Ã, the perturbation determinant Δ_Ã′/Ã(·) is the ratio of two ordinary determinants involving the Weyl function and boundary operators. In particular, if the deficiency indices of A are finite, then we obtain Δ_Ã′/Ã(z) = det (B′ - M(z))/det (B - M (z)), z ∈ ρ(Ã), where M(·) stands for the Weyl function and B′ and B for the boundary operators corresponding to Ã′ and à with respect to a chosen boundary triplet Π. The results are applied to ordinary differential operators and to second-order elliptic operators.