An abstract framework for interior-boundary conditions

Abstract

In a configuration space whose boundary can be identified with a subset of its interior, a boundary condition can relate the behaviour of a function on the boundary and in the interior. Additionally, boundary values can appear as additive perturbations. Such boundary conditions have recently provided insight into problems form quantum field theory. We discuss interior-boundary conditions in an abstract setting, with a focus on self-adjoint operators, proving self-adjointness criteria, resolvent formulas, and a classification theorem.

Hamiltonians for polaron models with subcritical ultraviolet singularities

“…Such conditions are known as interior-boundary condtions [30]. This can be related to the general theory of self-adjoint extensions [2,25]. The equality in point 3) can be interpreted as follows.…”

“…Conjugating each term in (1) with the lift Γ(U λ ) of this unitary to Fock space (acting as U λ on each tensor factor), we obtain Γ(U λ )HΓ(U λ ) * = λ γ Ω(dΓ(k) − λ −1 P ) + λ β dΓ(ω) + λ d/2−α (a(v) + a * (v)). (2) If we assume that γ = β (as we will later) and divide by λ γ , we thus find that λ −γ Γ(U λ )HΓ(U λ ) * = Ω(dΓ(k) − λ −1 P ) + dΓ(ω) + λ d/2−α−γ (a(v) + a * (v)) (3) is of the same form as (1) but with modified total momentum P λ = λ −1 P and coupling g λ = λ d/2−α−γ g. The large-momentum, respectively small-distance, behaviour of the model is related to the rescaled model with large λ. This has a small coupling constant if d/2 − α − γ < 0, and we call this scaling subcritical (d/2 − α − γ = 0, or > 0, would be critical and supercritical, respectively).…”

“…This method generalises the one used in [18,19] for the specific case of the Bogoliubov-Fröhlich Hamiltonian. The underlying idea is that one might be able to make sense of the expression (1) if one can find a domain D such that the action of the individual terms in H on Ψ ∈ D may not be an element of F, but, due to cancellations between the different terms, their sum is an element of F. The conditions on Ψ leading to such cancellations are known as (abstract) interior boundary conditions and have recently been studied for a variety of models [31,34,30,16,21,20,18,28,26,27,32,15,22,17,2].…”