A remark on the discriminant of Hill’s equation and Herglotz functions

Pankrashkin, Konstantin

doi:10.1007/s00013-014-0612-x

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“…This implies that the spectrum of H is composed of bands and gaps. More specifically, we have the following (see [Pan14] for a simple proof, or [RS78, Theorem XIII.89]).…”

Section: 5mentioning

confidence: 99%

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Edge states in ordinary differential equations for dislocations

Gontier

2020

Journal of Mathematical Physics

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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 = ...

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“…This implies that the spectrum of H is composed of bands and gaps. More specifically, we have the following (see [Pan14] for a simple proof, or [RS78, Theorem XIII.89]).…”

Section: 5mentioning

confidence: 99%

“…This implies that the spectrum of H is composed of bands and gaps. More specifically, we have the following (see [Pan14] for a simple proof, or [RS78, Theorem XIII.89]). Since V is 1-periodic, we have T E (x + n) = T n E T E (x), so the behaviour of the solutions at infinity depends on the singular values of T E .…”

Section: U (∂mentioning

confidence: 99%

Edge states in ordinary differential equations for dislocations

Gontier

2020

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

A remark on the discriminant of Hill’s equation and Herglotz functions

Cited by 1 publication

References 25 publications

Edge states in ordinary differential equations for dislocations

Edge states in ordinary differential equations for dislocations

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