We show here that the spectrum of the family of non-commutative harmonic oscillators Q_(α,β)(x, D) for α, β ∈ R+ in the range αβ = 1 is [0, + ∞) and is entirely essential spectrum. The previous existing results concern the case αβ > 1 (case in which Q_(α,β)(x, D) is globally elliptic with a discrete spectrum whose qualitative properties are being extensively studied), and ours therefore extend the picture to the range of parameters αβ ≥ 1.
Alberto Parmeggiani, Alberto Venni (2013). On the essential spectrum of certain non-commutative oscillators. JOURNAL OF MATHEMATICAL PHYSICS, 54(12), 121507-1-121507-10 [10.1063/1.4850876].
On the essential spectrum of certain non-commutative oscillators
PARMEGGIANI, ALBERTO;VENNI, ALBERTO
2013
Abstract
We show here that the spectrum of the family of non-commutative harmonic oscillators Q_(α,β)(x, D) for α, β ∈ R+ in the range αβ = 1 is [0, + ∞) and is entirely essential spectrum. The previous existing results concern the case αβ > 1 (case in which Q_(α,β)(x, D) is globally elliptic with a discrete spectrum whose qualitative properties are being extensively studied), and ours therefore extend the picture to the range of parameters αβ ≥ 1.File in questo prodotto:
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