Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces

Behrndt, Jussi; Exner, Pavel; Holzmann, Markus; Lotoreichik, Vladimir

doi:10.1002/mana.201500498

Cited by 37 publications

(50 citation statements)

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“…in the sense of distributions, where δ 0 denotes the Dirac measure at the origin. In [1] it is proved that ∆ + V → ∆ + ( V )δ 0 in the norm resolvent sense when → 0, and in [5] this result is generalized to higher dimensions for singular perturbations on general smooth hypersurfaces.…”

Section: Introductionmentioning

confidence: 94%

Klein’s paradox and the relativistic δ-shell interaction in ℝ3

Blesa¹,

Pizzichillo²

2018

Analysis & PDE

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Section: Introductionmentioning

confidence: 94%

Klein’s paradox and the relativistic δ-shell interaction in ℝ3

Blesa¹,

Pizzichillo²

2018

Analysis & PDE

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“…Choosing δ ∈ (0, 1) this implies the claim (B.2). 1 Note that this result is formulated in [6] only for C 2 -hypersurfaces but remains valid in the slightly less regular situation considered here. In fact, the key ingredient in the proof of [6, Proposition 3.1] that needs to be ensured for a regular, closed C Let us denote by κ =γ 2γ1 −γ 1γ2 the signed curvature of Σ, where γ = (γ 1 , γ 2 ) : I → R 2 is any natural parametrization of Σ (|γ| = 1).…”

Section: Appendix B Proof Of Theorem 45mentioning

confidence: 86%

“…The following theorem contains the result that H ε converges in the norm resolvent sense to A α ; we would like to point out that the interaction strength α of the limit operator is some suitable mean value of the potential V along the normal direction, see (4.9) below. Our proof uses a method which differs from the one in [6,31,32]. Since this proof is of more technical nature we postpone it to Appendix B. Theorem 4.5.…”

Section: Approximation Of a α By Landau Hamiltonians With Regular Potmentioning

confidence: 99%

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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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“…The conditions of such type appear if one considers singular potential supported on hypersurface. These potentials have been intensively investigated during last two decades (see, e.g., [21][22][23][24][25]).…”

Section: Asymptotics Constructionmentioning

confidence: 99%

On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier

Vorobiev¹,

Bagmutov²,

Popov³

2019

Nanosystems: Phys. Chem. Math.

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A quantum waveguide with a semitransparent barrier, placed across it, is considered. It is assumed that the barrier has a small window. This local perturbation of the waveguide causes the appearance of resonance states localized near the barrier with the window. The asymptotics (in small parameter -the window width) of the resonances (quasi-bound states) is obtained. The procedure of construction of full formal asymptotic expansion is described. The first two terms of the asymptotic expansion are obtained explicitly. These terms describe the shift of the resonance from the threshold and the life time of the corresponding resonance state.

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Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces

Cited by 37 publications

References 37 publications

Klein’s paradox and the relativistic δ-shell interaction in ℝ3

Klein’s paradox and the relativistic δ-shell interaction in ℝ3

The Landau Hamiltonian with δ-potentials supported on curves

On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier

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