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SPECTRAL BOUNDS FOR NON-SMOOTH PERTURBATIONS OF THE LANDAU HAMILTONIAN

Barbe, Jacques 大阪大学 DOI:10.18910/93060

2023.10

概要

the self-adjoint realization in L2 (R2 ) of the second order partial differential operator

2 
2
1 ∂
1 ∂
1
1
+ y +
− x
i ∂x 2
i ∂y 2
initially defined on (R2 ); this operator is usually called Landau Hamiltonian. The spectrum
of H0 consists of eigenvalues Λq := 2q + 1 of infinite multiplicity, the Landau levels (see
4.1).
Let V = Opw (b) be a bounded selfadjoint pseudo-differential operator (ΨDO). Under the
general assumption that V H0−1 is a compact operator, the Kato-Rellich theorem and the Weyl
perturbation theorem imply that the operator H0 − V is selfadjoint and that
σess (H0 − V) = σess (H0 ) = σ(H0 ) .
Hence the discrete spectrum of H0 − V consists of eigenvalues of finite multiplicity; these
eigenvalues can accumulate only to the Landau levels.
Suppose furthermore that the operator V is non-negative. Fix q ≥ 1 and let E  be a fixed
real number in the gap (Λq−1 , Λq ) and E a positive real number such that Λq−1 < E  <
Λq − E < Λq . ...

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8 all´ee des Favri`eres

44240, La Chapelle-sur-Erdre

France

e-mail: js.barbe2@wanadoo.fr

...

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