Edge states in ordinary differential equations for dislocations

Abstract

In this article, we study Schrödinger operators on the real line, when the external potential represents a dislocation in a periodic medium. We study how the spectrum varies with the dislocation parameter. We introduce several integer-valued indices, including Chern number for bulk indices, and various spectral flows for edge indices. We prove that all these indices coincide, providing a proof a bulk-edge correspondence in this case. The study is also made for dislocations in Dirac models on the real line. We prove that 0 is always an eigenvalue of such operators. arXiv:1908.01377v1 [math-ph] 4 Aug 2019Chern number. In Section 3.2, we focus on the operator H(t), and we consider the projector on the n lowest bands P n (t) := 1(H(t) ≤ E). Since P n (t) commutes with translations, we can Bloch transform it, and obtain a family of rank-n projectors P n (t, k) acting on L 2 ([0, 1]), which is periodic in both t and k. For such periodic family of projectors we associate a Chern number Ch(P n ) ∈ Z.Edge index. In Section 3.3, we see the (edge) equation H χ (t)u = Eu as a 1-periodic family of ODEs. We introduce the vectorial spaces L ,± χ (t) of solutions that decay at ±∞. These spaces may cross, and if u ∈ L ,+ χ (t) ∩ L ,− χ (t), then u is an eigenvector for H χ (t), that is an edge state. We associate to such bi-family of vectorial spaces an edge index I χ,n ∈ Z.Domain wall spectral flow. In Section 3.4, we focus on the edge operator H χ (t). Although its essential spectrum is independent of t, some eigenvalues may appear in the essential gaps. The spectral flow S χ,n ∈ Z is the net flow of eigenvalues going downwards through the gap.Dirichlet spectral flow. Finally, in Section 3.5, we consider the operator H D (t). Again, its essential spectrum is independent of t, and some eigenvalues may appear in its essential gaps. We associate a spectral flow S D,n to this family as well.Remark 1.1. As in [Dro18], we chose the convention to count the flow of eigenvalues going downwards for the spectral flow. This allows a nicer statement of the following theorem.The main result of this article can be summarised as follows. Theorem 1.2 (Bulk-edge correspondence). If the n-th gap is open, then B n = Ch(P n ) = I χ,n = S χ,n = S D,n = n.In particular, the indices are independent of χ and of the choice of E in the gap. In addition, all eigenvalues of H χ and H D are simple, and the corresponding eigenstates are exponentially localised.This result is more or less already known: the proof of Hatsugai [Hat93] in the discrete case can be used in our continuous setting to prove Ch(P n ) = S D,n . The equality B n = I D,n was proved in a discrete setting in [ASBVB13]. Combining the two results gives Ch(P n ) = B n , a fact noticed in [ASBVB13] (we also refer to the new articles [Bal17, Bal18] for a study in a continuous twodimensional setting). Recently, Drouot [Dro18] proved Ch(P n ) = S χ,n in the continuous setting, under the extra condition that the n-th gap does not close under a particular deformation. Finally, the equality S χ,n = ...

Edge states for second order elliptic operators

“…Our main result can be stated as follows (here for the Hill's case). It extends the one in the previous work [Gon20] and [Dro18]. Let n ∈ N \ {0} be fixed, and let…”

“…In the context of continuous models, it is unclear that one can define an index which is indeed independent of the chosen boundary conditions. In [Gon20], we proved that it was the case in a simple one-dimensional model for dislocations. We extend this work here, and give a general framework to define the edge index for different self-adjoint extensions of Schrödinger operators.…”

The bulk-edge correspondence for continuous dislocated systems

“…When the n-th gap is open, Korotyaev [72] and Dohnal-Plum-Reichel [25] show that there exist 0 = t 0 < t 1 < • • • < t n = 1 such that for every t ∈ (t j , t j+1 ), the operator Q(t) has a unique eigenvalue in the n-th gap. The effective flow of eigenvalues of Q(t) in the nth gap is n (see also Gonthier [47]). Hempel-Kohlmann [58] rediscovered this formula via a clever trick, flexible enough to work in two-dimensional analogs [57,60].…”

Edge states for second order elliptic operators in a channel

“…In the context of continuous models, it is unclear that one can define an index which is indeed independent of the chosen boundary conditions. In [20], we proved that it was the case in a simple one-dimensional model for dislocations. We extend this work here, and give a general framework to define the edge index for different self-adjoint extensions of Schrödinger operators.…”

“…When t is seen as the time variable, this equation models a Thouless pump [9,42]. In the case where V t .x/ D V .x t/, the variable t is interpreted as a dislocation parameter [16,20]. On the second part of the article, we study its PDE version, that is families of Schrödinger's operators of the form H t WD C V t .x; y/; acting on L 2 .R .0; 1/ d 1 ; C/:…”