Extreme Jensen measures

Abstract

Let Ω be an open subset of R d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure µ, supported on a compact subset of Ω, such that u dµ≤u(x) for every superharmonic function u on Ω. Denote by Jx(Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(Jx(Ω)), the set of extreme elements of Jx(Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29-53.As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {αn} ∞ n=1 and a continuous function M : C!R + , there exists an entire function f ≡0 satisfying f (αn)=0 for all n, and |f (z)|≤M (z) for all z ∈C.

Quantum Spin Hall Effect in Three Dimensional Materials: Lattice Computation of Z2 Topological Invariants and Its Application to Bi and Sb

“…[20][21][22][23] This makes it difficult to investigate realistic models, in which complicated many-band structure is involved. Therefore, for the direct study of Bi in 3D as well as for the search for other materials, to establish a simple and efficient computational method of Z 2 invariants in 3D is an urgent issue to be resolved.…”

“…24) The zeros of pðkÞ thus serve as an obstruction of the gauge fixing. 28) In systems with breaking T symmetry like QH effect, such an obstruction gives in general a nontrivial Chern number. Contrary to this, in systems under consideration, the Chern number always vanishes due to T invariance.…”

“…In 3D, it has been shown that the phases of T invariant systems are classified by four independent Z 2 invariants. [20][21][22] To compute them, let us define six two-dimensional tori, according to Moore and Balents. 21) For example, fix the third momentum to k 3 ¼ 0 or , then we have two tori spanned by k 1 and k 2 which we denote Z 0 and Z torus, respectively.…”

Classification of topological insulators and superconductors in three spatial dimensions

“…7,8,9,10,11,12,13,14,15,16 These topological states occur in certain materials with a bulk band gap generated by strong spin-orbit interactions and are known as Z 2 topological insulators. Unlike the integer quantum Hall states, the two-dimensional version of the Z 2 topological insulator, which has been dubbed "quantum spin Hall" (QSH) state, does not carry any net charge current along the edges.…”

Three-dimensional topological phase on the diamond lattice

“…1,2,3,4,5,6,7,8,9 On the one hand, Z 2 topological insulators are close relatives to more familiar integer quantum Hall (IQH) states. 10,11 As in an IQH state in the bulk, they are characterized by a topological invariant (Z 2 invariant).…”