Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

Abstract

Abstract. The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with δ and δ ′ -potentials supported on hypersurfaces.

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

“…We show that S coincides with the symmetric operator S in (9). Note first that Theorem 2.2 also implies that…”

“…Moreover, the spectral properties of A δ,α can be described with the help of the perturbation term (Θ δ,α − M (λ)) −1 . We mention that in the context of the more general notion of quasi boundary triples and their Weyl functions from [5,7] a similar approach as in this note and closely related results can be found in [8,9]; we also refer to [26,27,29,31,35,[38][39][40] for other methods in extension theory of elliptic differential operators.…”

“…Let S be the densely defined, closed, symmetric operator in (9). Then the adjoint S * of S is given by…”

Scattering of particles bounded to an infinite planar curve

“…However, [6] consider the case of boundary triples which are insufficient for the partial differential operators where quasi-boundary triples should be used (cf. discussions in the Introductions of [8,17]). The proof of Theorem A in [7] rely on compact embedding of H 1 (Γ(R)) into L 2 (Γ(R)] which does not hold for the infinite curve.…”

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.…”