Resolvent convergence in norm for Dirac operator with Aharonov–Bohm field

2003 ◽  
Vol 44 (7) ◽  
pp. 2967 ◽  
Author(s):  
Hideo Tamura
Keyword(s):  
Dirac Operator ◽  
Resolvent Convergence ◽  
Convergence In Norm ◽  
Aharonov Bohm

scholarly journals ON INDICES OF THE DIRAC OPERATOR IN A NON-FREDHOLM CASE

1996 ◽  
Vol 11 (12) ◽  
pp. 979-986
Author(s):  
ALEXANDER MOROZ
Keyword(s):  
Dirac Operator ◽  
Heat Kernel ◽  
Continuous Spectrum ◽  
Fredholm Operator ◽  
Axial Anomaly ◽  
Bohm Potential ◽  
Spectral Asymmetry ◽  
Aharonov Bohm ◽  
Dirac Hamiltonian

The Dirac–Hamiltonian with the Aharonov–Bohm potential provides an example of a non-Fredholm operator for which all spectral asymmetry comes entirely from the continuous spectrum. In this case one finds that the use of standard definitions of the resolvent regularized, the heat kernel regularized, and the Witten indices misses the contribution coming from the continuous spectrum and gives vanishing spectral asymmetry and axial anomaly. This behavior in the case of the continuous spectrum seems to be general and its origin is discussed.

Phenomena and devices at the quantum scale and the mesoscale

2017 ◽  
Author(s):  
Sandip Tiwari
Keyword(s):  
Coulomb Blockade ◽  
Secure Communication ◽  
Quantum Hall ◽  
Nanoscale Device ◽  
Self Energy ◽  
Stability Diagrams ◽  
Charge Stability ◽  
Bohm Effect ◽  
Aharonov Bohm ◽  
Non Locality

Unique nanoscale phenomena arise in quantum and mesoscale properties and there are additional intriguing twists from effects that are classical in origin. In this chapter, these are brought forth through an exploration of quantum computation with the important notions of superposition, entanglement, non-locality, cryptography and secure communication. The quantum mesoscale and implications of nonlocality of potential are discussed through Aharonov-Bohm effect, the quantum Hall effect in its various forms including spin, and these are unified through a topological discussion. Single electron effect as a classical phenomenon with Coulomb blockade including in multiple dot systems where charge stability diagrams may be drawn as phase diagram is discussed, and is also extended to explore the even-odd and Kondo consequences for quantum-dot transport. This brings up the self-energy discussion important to nanoscale device understanding.

scholarly journals Beating of Aharonov-Bohm oscillations in a closed-loop interferometer

2007 ◽  
Vol 76 (3) ◽  
Author(s):  
Sanghyun Jo ◽  
Gyong Luck Khym ◽  
Dong-In Chang ◽  
Yunchul Chung ◽  
Hu-Jong Lee ◽  
...  
Keyword(s):  
Closed Loop ◽  
Aharonov Bohm

scholarly journals Two-flux tunable Aharonov-Bohm effect in a photonic lattice

2021 ◽  
Vol 104 (2) ◽  
Author(s):  
V. Brosco ◽  
L. Pilozzi ◽  
C. Conti
Keyword(s):  
Bohm Effect ◽  
Aharonov Bohm

scholarly journals Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Maozhu Zhang ◽  
Kun Li ◽  
Hongxiang Song
Keyword(s):  
Boundary Conditions ◽  
Operator Theory ◽  
Resolvent Convergence ◽  
Spectral Inclusion ◽  
Regular Approximation ◽  
Special Cases ◽  
Abstract Operator ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

scholarly journals Extended supersymmetries and the Dirac operator

2005 ◽  
Vol 315 (2) ◽  
pp. 467-487 ◽  
Author(s):  
A. Kirchberg ◽  
J.D. Länge ◽  
A. Wipf
Keyword(s):  
Dirac Operator

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

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