Counting Function of Magnetic Resonances for Exterior Problems

Bruneau, Vincent; Sambou, Diomba

doi:10.1007/s00023-016-0497-2

Cited by 2 publications

(2 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For additive perturbations of A 0 by an electric potential this was shown by G. Raikov in [69], see also [38,54,62,66,70,74,75]. More recently similar results were proved in [20,19,44,64,67] for Landau Hamiltonians on domains with Dirichlet, Neumann, and Robin boundary conditions; for closely related results in three-dimensional situation we refer to [16,21] and the references therein.…”

Section: Introductionsupporting

confidence: 55%

The Landau Hamiltonian with δ-potentials supported on curves

et al. 2019

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionsupporting

confidence: 55%

The Landau Hamiltonian with δ-potentials supported on curves

et al. 2019

View full text Add to dashboard Cite

show abstract

“…A problem closely related to the analysis of the SSF ξ(•; H 0 + V, H 0 ) as E → Λ q for a given q ∈ Z + , is the investigation of accumulation of resonances of H 0 + V at Λ q performed in [8,9,10]. The asymptotic distribution of resonances near the Landau levels for the operators H ± considered in this article, is studied in [12]. Let us mention also some 2D results related to Theorem 3.1.…”

Section: Resultsmentioning

confidence: 99%

Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

Bruneau¹,

Raikov²

2020

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

We consider the 3D Schrödinger operator H 0 with constant magnetic field B of scalar intensity b > 0, and its perturbations H + (resp., H − ) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain Ω in ⊂ R 3 . We introduce the Krein spectral shift functions ξ(E; H ± , H 0 ), E ≥ 0, for the operator pairs (H ± , H 0 ), and study their singularities at the Landau levels Λ q := b(2q + 1), q ∈ Z + , which play the role of thresholds in the spectrum of H 0 . We show that ξ(E; H + , H 0 ) remains bounded as E ↑ Λ q , q ∈ Z + , being fixed, and obtain three asymptotic terms of ξ(E; H − , H 0 ) as E ↑ Λ q , and of ξ(E; H ± , H 0 ) as E ↓ Λ q . The first two terms are independent of the perturbation while the third one involves the logarithmic capacity of the projection of Ω in onto the plane perpendicular to B.

show abstract

Counting Function of Magnetic Resonances for Exterior Problems

Cited by 2 publications

References 22 publications

The Landau Hamiltonian with δ-potentials supported on curves

The Landau Hamiltonian with δ-potentials supported on curves

Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

Contact Info

Product

Resources

About