Abstract
The main purpose of this note is to study spectral properties of the Stark magnetic Hamiltonian : , on the Hilbert space L 2 (R 2 ). We show that if the potential V satisfies some mild regularity conditions and is sufficiently decaying at infinity, then the operator H(μ, ϵ) has possibly at most a finite number of eigenvalues.
Highlights
The two-dimensional Schrodinger operator with constant electric and magnetic fields can be written as : H(μ, ε) = H0(μ, ε) + V (x, y), whereH0(μ, ε) = (Dx − μy)2 + Dy2 + εx, Dν = −i∂ν .and μ and ε are proportional to the strength of the corresponding homogeneous magnetic and electric fields
Μ and ε are proportional to the strength of the corresponding homogeneous magnetic and electric fields
The perturbation V creates discrete eigenvalues which can be accumulated near the Landau levels
Summary
The numbers λn = (2n + 1)μ, n = 0, 1, 2, · · · are called the Landau levels and they are eigenvalues of the magnetic hamiltonian H0(μ, 0), each of them with infinite multiplicity. The asymptotic behavior of the function counting the number of eigenvalues of H(μ, 0), in a neighborhood of a Landau level λn, has been studied by many authors in different aspects (see [1,3,4, 9,10]). It was shown in [9, 10] that the number of eigenvalues of H(μ, 0) in ]λn − η, λn[ ∪]λn, λn + δ[ is infinite, ∀η > 0, δ > 0, even if V is compactly supported.
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