skip to main content

SciTech ConnectSciTech Connect

Title: A quantum Cherry theorem for perturbations of the plane rotator

We consider on L{sup 2}(T{sup 2}) the Schrödinger operator family L{sub ε}:ε∈R with domain and action defined as D(L{sub ε})=H{sup 2}(T{sup 2}), L{sub ε}u=−(1/2)ℏ{sup 2}(α{sub 1}∂{sub φ{sub 1}{sup 2}}+α{sub 2}∂{sub φ{sub 2}{sup 2}})u−iℏ(γ{sub 1}∂{sub φ{sub 1}}+γ{sub 2}∂{sub φ{sub 2}})u+εVu. Here ε∈R, α= (α{sub 1}, α{sub 2}), γ= (γ{sub 1}, γ{sub 2}) are vectors of complex non-real frequencies, and V a pseudodifferential operator of order zero. L{sub ε} represents the Weyl quantization of the Hamiltonian family L{sub ε}(ξ,x)=(1/2)(α{sub 1}ξ{sub 1}{sup 2}+α{sub 2}ξ{sub 2}{sup 2})+γ{sub 1}ξ{sub 1}+γ{sub 2}ξ{sub 2}+εV(ξ,x) defined on the phase space R{sup 2}×T{sup 2}, where V(ξ,x)∈C{sup 2}(R{sup 2}×T{sup 2};R). We prove the uniform convergence with respect to ℏ∈[0, 1] of the quantum normal form, which reduces to the classical one for ℏ= 0. This result simultaneously entails an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical Birkhoff normal form generalizing a well known theorem of Cherry.
Authors:
 [1] ;  [2]
  1. Dipartimento di Matematica, Università di Bari, 70122 Bari (Italy)
  2. Dipartimento di Matematica, Università di Bologna, 40127 Bologna (Italy)
Publication Date:
OSTI Identifier:
22217739
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 12; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; HAMILTONIANS; PERTURBATION THEORY; PHASE SPACE; QUANTIZATION; SCHROEDINGER EQUATION; VECTORS