A variational approach to dislocation problems for periodic Schrödinger operators

Hempel, Rainer; Kohlmann, Martin

doi:10.1016/j.jmaa.2011.03.050

Cited by 23 publications

(40 citation statements)

References 16 publications

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“…We now give a condensed account of some of the results in [12] concerning the operators S t and D t . We begin with S t where we first note the following well-known basic facts.…”

Section: The Dislocation Problem On a Strip And For The Planementioning

confidence: 99%

“…Some results on the surface i.d.s. measure for the translational dislocation problem can be found in [12].…”

Section: Integrated Density Of States Boundsmentioning

confidence: 99%

“…A different approach was most recently implemented in [12] where it is shown that some of the results of [16] and [17] can be recovered with a rather crude variational technique. This method can be easily generalized to the dislocation problem on a strip R .0; 1/ and then to the plane R 2 ; one finds that spectrum of D t crosses the gap as t varies between 0 and 1.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we briefly summarize some results of [12] on translational lattice dislocations for the strip and for the plane. What we will use in the sequel is the simple fact that, for any E 2 .a; b/, there is some t D t E 2 .0; 1/ with the following property: for any " > 0, there is a compactly supported approximate eigenfunction u in the domain of D t that satisfies jj.D t E/ujj < " and jjujj D 1.…”

Section: Introductionmentioning

confidence: 99%

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Spectral properties of grain boundaries at small angles of rotation

Hempel¹,

Kohlmann²

2011

J. Spectr. Theory

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Abstract. We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential V W R 2 ! R, we let V # .x; y/ D V .x; y/ in the right half-plane fx 0g and V # D V B M # in the left half-plane fx < 0g, where M # 2 R 2 2 is the usual matrix describing rotation of the coordinates in R 2 by an angle #. As a main result, it is shown that spectral gaps of the periodic Schrödinger operator H D C V fill with spectrum of R # D C V # as 0 ¤ # ! 0. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states. Mathematics Subject Classification (2010). Primary 35J10; Secondary 35P20, 81Q10.

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“…We now give a condensed account of some of the results in [12] concerning the operators S t and D t . We begin with S t where we first note the following well-known basic facts.…”

Section: The Dislocation Problem On a Strip And For The Planementioning

confidence: 99%

“…Some results on the surface i.d.s. measure for the translational dislocation problem can be found in [12].…”

Section: Integrated Density Of States Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral properties of grain boundaries at small angles of rotation

Hempel¹,

Kohlmann²

2011

J. Spectr. Theory

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“…This paper is related to a mechanism for the emergence of defect states due to perturbations which are not spatially localized. Such perturbations may arise in models of dislocations in crystals; see, for example, [30,37].…”

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confidence: 99%

Dirac Operators and Domain Walls

Lu¹,

Watson²,

Weinstein³

2020

SIAM J. Math. Anal.

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We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying "mass" term. It is well-known that when the mass function has the form of a kink, or domain wall, transitioning between strictly positive and strictly negative asymptotic mass, ±κ∞, at ±∞, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding exponentially localized zero mode.We consider the eigenvalue problem for the one-dimensional Dirac operator with mass function defined by "glue-ing" together n domain wall-type transitions, assuming that the distance between transitions, 2δ, is sufficiently large, focusing on the illustrative cases n = 2 and 3. When n = 2 we prove that the Dirac operator has two real simple eigenvalues of opposite sign and of order e −2|κ∞|δ . The associated eigenfunctions are, up to L 2 error of order e −2|κ∞|δ , linear combinations of shifted copies of the single domain wall zero mode. For the case n = 3, we prove the Dirac operator has two non-zero simple eigenvalues as in the two domain wall case and a simple eigenvalue at energy zero. The associated eigenfunctions of these eigenvalues can again, up to small error, be expressed as linear combinations of shifted copies of the single domain wall zero mode. When n > 3 no new technical difficulty arises and the result is similar. Our methods are based on a Lyapunov-Schmidt reduction/Schur complement strategy, which maps the Dirac operator eigenvalue problem for eigenstates with near-zero energies to the problem of determining the kernel of an n × n matrix reduction, which depends nonlinearly on the eigenvalue parameter.The class of Dirac operators we consider controls the bifurcation of topologically protected "edge states" from Dirac points (linear band crossings) for classes of Schrödinger operators with domain-wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.

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Dislocation Problems for Periodic Schrödinger Operators and Mathematical Aspects of Small Angle Grain Boundaries

Hempel

Kohlmann

2012

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

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Abstract. We discuss two types of defects in two-dimensional lattices, namely (1) translational dislocations and (2) defects produced by a rotation of the lattice in a half-space.For Lipschitz-continuous and Z 2 -periodic potentials, we first show that translational dislocations produce spectrum inside the gaps of the periodic problem; we also give estimates for the (integrated) density of the associated surface states. We then study lattices with a small angle defect where we find that the gaps of the periodic problem fill with spectrum as the defect angle goes to zero. To introduce our methods, we begin with the study of dislocation problems on the real line and on an infinite strip. Finally, we consider examples of muffin tin type. Our overview refers to results in [HK1, HK2].

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A variational approach to dislocation problems for periodic Schrödinger operators

Cited by 23 publications

References 16 publications

Spectral properties of grain boundaries at small angles of rotation

Spectral properties of grain boundaries at small angles of rotation

Dirac Operators and Domain Walls

Dislocation Problems for Periodic Schrödinger Operators and Mathematical Aspects of Small Angle Grain Boundaries

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