Absence of the singular continuous component in spectra of analytic direct integrals

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potentialshown that the Schrödinger operators H t = − + W t have spectrum (surface states) in the spectral gaps of H 0 , for suitable t ∈ (0, 1). We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem (Korotyaev (2000(Korotyaev ( , 2005 [15,16]) on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In Appendix A, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.

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“…[6,11]. As for the spectrum of S t inside the gaps of S 0 , Theorem 4.1 leads to the following result.…”

Section: (S N0 − E)[ψ N/4 U N ] ∼ = (S Nt N − E)(ψ N/4 U N )mentioning

confidence: 86%

A variational approach to dislocation problems for periodic Schrödinger operators

Hempel

Kohlmann

2011

Journal of Mathematical Analysis and Applications

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“…Assuming either (4) Remark 2.1. With these results in hand we may conclude that the singular continuous component in the spectrum of H is empty [20].…”

Section: Absolute Continuity Of the Hamiltonianmentioning

confidence: 80%

On an extension of the Iwatsuka model

Tušek¹

2016

J. Phys. A: Math. Theor.

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“…We let (N,x) [1,2]). The singular spectrum of the operator D + W is empty and the eigenvalues (if they exist) have an infinite multiplicity and form a discrete set (see [3] and also [4], the last statement can also be easily derived from results of [5]). If there are no eigenvalues in the spectrum of the operator D + W , then the spectrum is absolutely continuous.…”

Section: Introductionmentioning

confidence: 97%

On Absolute Continuity of the Spectrum of a 3D Periodic Magnetic Dirac Operator

Danilov

2011

Integr. Equ. Oper. Theory

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Absolute continuity of the spectrum of a 3D periodic magnetic Dirac operator is proved provided that the magnetic potential A belongs to the space H q loc , q > 1, and the matrix potential V ∈ L 3 loc is represented in the form V = V0 + V1, where V0 commutes and V1 anticommutes with the Dirac matrices αj, j = 1, 2, 3. Mathematics Subject Classification (2010). Primary 35P05, 35Q40; Secondary 81Q10.Keywords. Dirac operator, periodic potential, absolutely continuous spectrum. IntroductionLet M M , M ∈ N, be the linear space of complex (M × M )-matrices, S M the set of Hermitian matrices from M M . Matrices α j ∈ S M , j = 1, . . . , d (d 2), are assumed to satisfy the anticommutation relations α j α l + α l α j = 2δ jl I, where I ∈ M M is the identity matrix and δ jl is the Kronecker delta. Let S M be the subset of S M that consists of matrices L represented in the form L = L 0 + L 1 , where L s ∈ S M and α j L s = (−1) s L s α j for all j = 1, . . . , d, s = 0, 1. Consider the d-dimensional Dirac operator d j=1The matrix function V : R d → S M and the components A j of the magnetic potential A : R d → R d are supposed to be periodic with a common period lattice Λ ⊂ R d . In particular, the matrix function V can be chosen in the form

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Absence of the singular continuous component in spectra of analytic direct integrals

Cited by 21 publications

References 3 publications

A variational approach to dislocation problems for periodic Schrödinger operators

A variational approach to dislocation problems for periodic Schrödinger operators

On an extension of the Iwatsuka model

On Absolute Continuity of the Spectrum of a 3D Periodic Magnetic Dirac Operator

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