Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

Abstract

Abstract. We investigate Schrödinger operators with δ and δ ′ -interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schrödinger operators with δ and δ ′ -interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ ′ -interactions.

Generalized interactions supported on hypersurfaces

“…Furthermore, for interactions supported on a noncompact, asymptotically planar hypersurface in R 3 we obtain a lower bound for the essential spectrum. In Section 4 we derive operator inequalities between different elements of the considered operator family, generalizing those between the δ-and δ ′ -interactions found in [4] and use them to establish further spectral results.…”

Dirac operators, shell interactions, and discontinuous gauge functions across the boundary

“…That is, we omit the underlying measures µ and σ associated to G and g, respectively, in the distributional relations stated in (5). In particular, we have Φ(X ) ⊂ L 2 (µ) 4 and 4 .…”

“…, where V is as in (5). (6) The following theorem, which is a direct application of [2, Theorem 2.11(iii)], shows that T Λ given by (6) is self-adjoint under some assumptions on Λ.…”

“…Since λ ∂Ω + M > 0 in ∂Ω, from Proposition 5.1 we deduce that T Λ λ ∂Ω +M is self-adjoint and T Λ λ ∂Ω +M = H + V λ ∂Ω +M on D T Λ λ ∂Ω +M , where V λ ∂Ω +M is given by (27). In particular, V λ ∂Ω +M = V on D T Λ λ ∂Ω +M , for V as in (5). Furthermore, thanks to Lemma 2.2(i) this means that if ϕ = Φ(G + g) ∈ D T Λ λ ∂Ω +M then i(α · N )(ϕ − − ϕ + ) = −g = V ϕ = V λ ∂Ω +M ϕ = (λ ∂Ω + M )(α · N ) ϕ + + ϕ − 2 , and therefore, using that λ ∂Ω + M is real-valued, we get (33)…”

Laplacians with singular perturbations supported on hypersurfaces

“…This is a relevant point for the interface perspective of studying the scattering problem. Moreover, while some sub-families of extensions (mainly those concerned with the δ or δ interface conditions) have been largely investigated by using quadratic form or quasi-boundary triple techniques (see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]), for others models presented in [1], and in particular those concerned with local interface conditions of Dirichlet and Neumann type, a rigorous analysis was not previously given.…”