Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

Bruneau, Vincent; Miranda, Pablo

doi:10.1016/j.jfa.2017.10.007

Cited by 7 publications

(13 citation statements)

References 38 publications

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“…It is obvious that under conditions of Theorem 4.1, the function N n pλq satisfies the same boundedness and continuity properties, wherever it is defined. Similar results for different magnetic Hamiltonians have been obtained before, see for example [6,3,4].…”

Section: The Spectral Shift Functionsupporting

confidence: 89%

“…Remark 4.5. Equation (45) is similar to the results obtained in [4] for the SSF of some magnetic Schrödinger operators defined in a Half-plane, and in [27], [23] for the eigenvalue counting function of the perturbed Landau Hamiltonian and Iwatsuka Hamiltonian, respectively.…”

Section: The Spectral Shift Functionsupporting

confidence: 82%

“…However, in the aforementioned articles it was not necessary to make a detailed analysis of the corresponding band functions; they are constants in the Landau case and some variational estimates are enough for the Iwatsuka case. There are other models where the asymptotics of band functions was necessary to make the spectral analysis, the closest one being a half-plane submitted to a constant magnetic field ( [5,17,4]). One of the differences with our situation is that we consider a whole class of varying magnetic field, and in these references the magnetic field is constant, allowing, for instance, the use of special functions to study the fiber operator.…”

Section: 2mentioning

confidence: 99%

“…Remark 4.6. Using the results of [17] combined with the methods of [4], it should be possible to prove (45) for a magnetic field satisfying bpxq " b`for x ą 0, and bpxq " b´for x ă 0. Note that this magnetic field does not satisfy (10) nor (5).…”

Section: The Spectral Shift Functionmentioning

confidence: 99%

“…The proofs of Theorems 4.1, 4.3 and 4.4 appear in the following two sections. They are inspired by [4] and [3].…”

Section: The Spectral Shift Functionmentioning

confidence: 99%

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Spectrum of the Iwatsuka Hamiltonian at thresholds

Miranda

Popoff

2018

Journal of Mathematical Analysis and Applications

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ABSTRACT. We consider the bi-dimensional Schrödinger operator with unidirectionally constant magnetic field, H 0 , sometimes known as the "Iwatsuka Hamiltonian". This operator is analytically fibered, with band functions converging to finite limits at infinity. We first obtain the asymptotic behavior of the band functions and its derivatives. Using this results we give estimates on the current and on the localization of states whose energy value is close to a given threshold in the spectrum of H 0 . In addition, for non-negative electric perturbations V we study the spectral density of H 0˘V , by considering the Spectral Shift Function associated to the operator pair pH 0V , H 0 q. We compute the asymptotic behavior of the Spectral Shift Function at the thresholds, which are the only points where it can grows to infinity.

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Section: The Spectral Shift Functionsupporting

confidence: 89%

Section: The Spectral Shift Functionsupporting

confidence: 82%

Section: 2mentioning

confidence: 99%

Section: The Spectral Shift Functionmentioning

confidence: 99%

“…The proofs of Theorems 4.1, 4.3 and 4.4 appear in the following two sections. They are inspired by [4] and [3].…”

Section: The Spectral Shift Functionmentioning

confidence: 99%

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Spectrum of the Iwatsuka Hamiltonian at thresholds

Miranda

Popoff

2018

Journal of Mathematical Analysis and Applications

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Spectral asymptotics at thresholds for a Dirac-type operator on Z2

Miranda

Parra

Raikov

2023

Journal of Functional Analysis

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The Landau Hamiltonian with δ-potentials supported on curves

et al. 2019

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The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian Aα = (i∇ + A) 2 + αδΣ in L 2 (R 2 ) with a δ-potential supported on a finite C 1,1 -smooth curve Σ are studied. Here A = 1 2 B(−x2, x1) ⊤ is the vector potential, B > 0 is the strength of the homogeneous magnetic field, and α ∈ L ∞ (Σ) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Σ. After a general discussion of the qualitative spectral properties of Aα and its resolvent, one of the main objectives in the present paper is a local spectral analysis of Aα near the Landau levels B(2q + 1), q ∈ N0. Under various conditions on α it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of α. Furthermore, the use of Landau Hamiltonians with δ-perturbations as model operators for more realistic quantum systems is justified by showing that Aα can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.Organization of the paper. Section 2 contains some preliminary material concerning the unperturbed Landau Hamiltonian, properties of Schatten-von Neumann ideals and some aspects of perturbation theory. In Subsection 2.4 we discuss a class of Toeplitz-like operators related to Landau Hamiltonians. In Section 3 we make use of the abstract concept of quasi boundary triples and their Weyl functions (see Appendix A for a brief introduction) in order to study Landau Hamiltonians with δ-potentials supported on curves. Using a suitable quasi boundary triple we show self-adjointness of A α , provide qualitative spectral properties, and derive the Krein-type resolvent formula (1.4). The approximation of A α by magnetic Schrödinger operators with scaled regular potentials is also discussed; the proof is technical and therefore outsourced to Appendix B. Section 5 is devoted to the spectral analysis of the compressed resolvent difference P q W λ P q . Under various assumptions we obtain spectral estimates and spectral asymptotics for this operator. Based on these results we provide our main results on the eigenvalue clusters of A α at Landau levels in Section 6.

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Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

Cited by 7 publications

References 38 publications

Spectrum of the Iwatsuka Hamiltonian at thresholds

Spectrum of the Iwatsuka Hamiltonian at thresholds

Spectral asymptotics at thresholds for a Dirac-type operator on Z2

The Landau Hamiltonian with δ-potentials supported on curves

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