The approximation of the Schrödinger operators with penetrable wall potentials in terms of short range Hamiltonians

Shimada, Shin-ichi

doi:10.1215/kjm/1250519494

Cited by 9 publications

(7 citation statements)

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“…Therefore, we get where b 2 is a constant independent of e(0 < e ^ 2" 1 ). Combining (3)(4)(5)(6)(7)(8)(9), (3)(4)(5)(6)(7)(8)(9)(10) and (3)(4)(5)(6)(7)(8)(9)(10)(11) and taking into account that MT is a bounded operator in H 1^2 ) and that ©(i? 2 ) is dense in /^(i?…”

Section: The Model Of Semi-transparent Surface and Short-range Potentialmentioning

confidence: 99%

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The operator extension theory, semitransparent surface and short range potential

Popov¹

1995

Math. Proc. Camb. Phil. Soc.

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Section: The Model Of Semi-transparent Surface and Short-range Potentialmentioning

confidence: 99%

“…It should be mentioned that there exist other interesting approaches to this problem [2,3,11] which have some common features with our model. In the process of justification of our model we use some ideas of proofs from [11].…”

Section: Introductionmentioning

confidence: 99%

The operator extension theory, semitransparent surface and short range potential

Popov¹

1995

Math. Proc. Camb. Phil. Soc.

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“…Hamiltonian for more realistic model operators with regular potentials. The approximation problem of singular potentials by regular ones has been discussed in the absence of magnetic fields for δ-point interactions in great detail in the monograph [4], and for δ-surface interactions in [6,31,32] and [18,41,63,65,77], see also [5,79] for more abstract approaches. We show in Theorem 4.5 and Corollary 4.6 that for real α ∈ L ∞ (Σ) the singular Landau Hamiltonian A α can be approximated in the norm resolvent sense by a family of regular Landau Hamiltonians with potentials suitably scaled in the direction perpendicular to Σ.…”

Section: Introductionmentioning

confidence: 99%

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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“…Apart from that, the literature on the approximation of Schrödinger operators with δ‐potentials supported on curves in

R^{2}

and surfaces in

R^{3}

is less complete; there are results available for the special cases that Σ is a sphere in

{double-struckR}^{3}

, , that Σ is the boundary of a star‐shaped domain in the plane , and that Σ is a smooth planar curve or surface and the interaction strength is constant , . In all of the above mentioned works convergence in the norm resolvent sense is shown.…”

Section: Introductionmentioning

confidence: 99%

Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces

et al. 2016

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We show that a Schrödinger operator Aδ,α with a δ‐interaction of strength α supported on a bounded or unbounded C2‐hypersurface Σ⊂double-struckRd,d≥2, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator Aδ,α with a singular interaction is regarded as a self‐adjoint realization of the formal differential expression −Δ−αfalse⟨δΣ,·false⟩δnormalΣ, where α:Σ→R is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.

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The approximation of the Schrödinger operators with penetrable wall potentials in terms of short range Hamiltonians

Cited by 9 publications

References 0 publications

The operator extension theory, semitransparent surface and short range potential

The operator extension theory, semitransparent surface and short range potential

The Landau Hamiltonian with δ-potentials supported on curves

Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces

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