A remark on Krein’s resolvent formula and boundary conditions

We prove an analog of Krein’s resolvent formula expressing the resolvents of self-adjoint extensions in terms of boundary conditions. Applications to quantum graphs and systems with point interactions are discussed. AMS classification scheme numbers: 46N50, 47A06, 47A10 PACS numbers: 02.30.Tb, 02.60.Lj Krein’s resolvent formula [1] is a powerful tool in the spectral analysis of self-adjoint extensions, which found numerous applications in many areas of mathematics and physics, including the study of exactly solvable models in quantum physics [2, 3, 4]. For the use of this formula in the traditional way one needs a kind of preliminary construction, like finding a maximal common part of two extensions, see [2, Appendix A]. While this is enough for many applications, including models with point interactions, there is a number of problems like the study of quantum graphs or more general hybrid structures, where self-adjoint extensions are suitable described by more complicated boundary conditions, see [5, 6, 7], and it is necessary to modify Krein’s resolvent formula to take into account these new needs. This can be done if one either modifies the coordinates in which the boundary data are calculated [6, 8] or considers boundary conditions given in a non-operator way using linear relations [9, 10]. On the other hand, a more attractive idea is to have a resolvent formula taking directly the boundary conditions into account, without changing the coordinates. We describe the realization of this idea in the present note. Let S be a closed densely defined symmetric operator with the deficiency indices (n,n), 0 < n < ∞, acting in a certain Hilbert space H . One says that a triple (V,Γ1,Γ2), where V = Cn and Γ1 and Γ2 are linear maps from the domain domS∗ of the adjoint of S to V , is a boundary value space for S if 〈φ ,S∗ψ〉− 〈S∗φ ,ψ〉 = 〈Γ1φ ,Γ2ψ〉− 〈Γ2φ ,Γ2ψ〉 for any φ ,ψ ∈ domS∗ and the map (Γ1,Γ2) : domS∗ →V ⊕V is surjective. A boundary value space always exists [9, Theorem 3.1.5]. All self-adjoint extensions of S are restrictions of S∗ to functions φ ∈ domS∗ satisfying AΓ1φ = BΓ2φ , where the matrices A and B must obey the following two properties: AB∗ = BA∗ ⇔ AB∗ is self-adjoint, (1) the n×2n matrix (A|B) has maximal rank n. (2)