Generalized interactions supported on hypersurfaces

Abstract

Abstract. We analyze a family of singular Schrödinger operators with local singular interactions supported by a hypersurface Σ ⊂ R n , n ≥ 2, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of Σ the interaction is characterized by four real parameters, the earlier studied case of δ-and δ ′ -interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

Asymptotic completeness and S-matrix for singular perturbations

“…The construction of the operator ∆ θ with semi-transparent boundary condition of δ ′ -type provided in Theorem 5.15 extend, as regards the regularity of the boundary and/or the class of admissible strength functions, previous constructions given in [6], [26], [15]. Asymptotic completeness for the scattering couple (∆, ∆ θ ) provided in Theorem 5.15 extend results on existence and completeness given, whenever the boundary is smooth and θ is bounded, in [6] and [26].…”

Scattering from local deformations of a semitransparent plane

“…In this section we give a rigorous definition of the operators A 0 and A F and obtain a Kreȋn's type formula for their resolvents. We remark that a rigorous definition of the operators A 0 and A F can also be obtained, more directly, starting from the associated quadratic form, see, e.g., [4,9,18]. However, since we shall extensively use Kreȋn's type resolvent formulae, we prefer to use them also to characterize the domain and action of the operators A 0 and A F .…”

“…the resolvent of A ∅ ) (see Theorem 3.4). Let us remark here that the operator A F could be equivalently defined by quadratic form methods (see, e.g., [4,9,18] and references therein); however, in order to study the Limiting Absorption Principle (LAP for short) and the Scattering Matrix, one needs a convenient resolvent formula. One more important remark about our use of resolvent formulae is the following: proceeding as in [15], one could try to provide a resolvent formula of A F expressed directly in terms of A 0 ; however this would imply the use of trace (evaluation) operators in the operator domain of A 0 , and these are less well-behaved than in the Sobolev space H 2 (R 3 ), the self-adjointness domain of A ∅ (in particular is not clear what should be the correct trace space).…”

Spectral isoperimetric inequalities for singular interactions on open arcs

“…The same technique can be used to reprove the optimization result in [EHL06] on δ-interactions without making use of the Birman-Schwinger principle. In fact, the method seems to be applicable for a larger sub-class of general fourparametric boundary conditions, considered in [ER16]. One has only to ensure that the lowest spectral point is indeed a negative eigenvalue and that the corresponding ground-state is real-valued and radially symmetric for the case of the interaction supported on a circle.…”