Interpolation Between Hilbert Spaces

Abstract

This note comprises a synthesis of certain results in the theory of exact interpolation between Hilbert spaces. In particular, we discuss the characterizations of all interpolation spaces [2] and of all quadratic interpolation spaces [13], and we give connections to other results in the area. Interpolation theoretic notions1.1. Interpolation norms. When X, Y are normed spaces, we use the symbol L(X; Y ) to denote the totality of bounded linear maps T : X → Y with the operator normWhen X = Y we simply write L(X). Consider a pair of Hilbert (or even Banach-) spaces H = (H 0 , H 1 ) which is regular in the sense that H 0 ∩ H 1 is dense in H 0 as well as in H 1 . We assume that the pair is compatible in the sense that H i ⊂ M for i = 0, 1 where M is some Hausdorff topological vector space.We define the K-functional ( 1 ) for the couple H by

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

“…[7,17,45]). In particular, the main result of [17] proves self-adjointness for Dirac operators on metric graphs with a wide family of linear vertex conditions (including the Kirchoff-type ones (8)- (9)).…”

“…Unfortunately, this definition is not the most suitable for the purposes of the paper. An alternative way to define the form domain of D (that is, dom(Q D )) is to use the well known Real Interpolation Theory [5,9]. Here we just mention some basics, referring to Appendix B for some further details.…”

“…It is then left to prove that ψ fulfills (8) and (9). First, fix a vertex v of the compact core and choose…”

“…(iii) prove that the sequence (u n ) satisfies (60), as by Lemmas 4.3 and 4.5, this entails that it is a Palais-Smale sequence for J; (iv) prove that the sequence (u n ) converges (up to subsequences) in H to a function u, which is then a bound state of the NLSE with frequency λ < 0. We observe that we always tacitly use in the following the fact that, since each ψ n is a bound state of the NLDE, then u n ∈ H, whereas v n ∈ H, but satisfies (9). In addition, we highlight that, in the sequel, we often use a "formal" commutation between the differential operator (·) ′ and χ K .…”

“…Roughly speaking these conditions "split" the requirements of Kirchhoff conditions: the continuity condition is imposed only on the first component of the spinor, while the second component (in place of the derivative) has to satisfy a "balance" condition (see (8)& (9)).…”