Eigenvalues and eigenfunctions for the two dimensional Schrödinger operator with strong magnetic field

2020 ◽  
Vol 120 (1-2) ◽  
pp. 175-197
Author(s):  
Naoya Yoshida
Keyword(s):  
Magnetic Field ◽  
Schrödinger Operator ◽  
Scalar Potential ◽  
Energy Curve ◽  
Effective Hamiltonian ◽  
Two Dimensional ◽  
Schrodinger Operator ◽  
Eigenvalues And Eigenfunctions ◽  
Pseudodifferential Calculus ◽  
Distribution Of Eigenvalues

We study the eigenvalues of the two-dimensional Schrödinger operator with a large constant magnetic field perturbed by a decaying scalar potential. For each Landau level, we give the precise asymptotic distribution of eigenvalues created by the minimum, maximum and the closed energy curve of the potential. Normal form reduction, WKB construction and pseudodifferential calculus are applied to the effective Hamiltonian.

scholarly journals On the spectrum and eigenfunctions of the Schrödinger operator with Aharonov-Bohm magnetic field

2005 ◽  
Vol 2005 (23) ◽  
pp. 3751-3766 ◽  
Author(s):  
Anders M. Hansson
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Schrödinger Operator ◽  
Scalar Potential ◽  
Fractional Part ◽  
Schrodinger Operator ◽  
Sharp Constants ◽  
Quadratic Potential ◽  
Aharonov Bohm ◽  
Classical Constant

We explicitly compute the spectrum and eigenfunctions of the magnetic Schrödinger operatorH(A→,V)=(i∇+A→)2+VinL2(ℝ2), with Aharonov-Bohm vector potential,A→(x1,x2)=α(−x2,x1)/|x|2, and either quadratic or Coulomb scalar potentialV. We also determine sharp constants in the CLR inequality, both dependent on the fractional part ofαand both greater than unity. In the case of quadratic potential, it turns out that the LT inequality holds for allγ≥1with the classical constant, as expected from the nonmagnetic system (harmonic oscillator).

scholarly journals On the two-dimensional Schrödinger operator with an attractive potential of the Bessel-Macdonald type

2020 ◽  
pp. 168385
Author(s):  
Wellisson B. De Lima ◽  
Oswaldo M. Del Cima ◽  
Daniel H.T. Franco ◽  
Bruno C. Neves
Keyword(s):  
Schrödinger Operator ◽  
Attractive Potential ◽  
Two Dimensional ◽  
Schrodinger Operator
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