Convergence of a quantum normal form and an exact quantization formula

@article{Graffi2011ConvergenceOA,
  title={Convergence of a quantum normal form and an exact quantization formula},
  author={S. Graffi and T. Paul},
  journal={Journal of Functional Analysis},
  year={2011},
  volume={262},
  pages={3340-3393}
}
  • S. Graffi, T. Paul
  • Published 2011
  • Mathematics, Physics
  • Journal of Functional Analysis
Abstract The operator − i ℏ ω ⋅ ∇ on L 2 ( T l ) , quantizing the linear flow of diophantine frequencies ω = ( ω 1 , … , ω l ) over T l , l > 1 , is perturbed by the operator quantizing a function V ω ( ξ , x ) = V ( ω ⋅ ξ , x ) : R l × T l → R , z ↦ V ( z , x ) : R × T l → R real-holomorphic. The corresponding quantum normal form (QNF) is proved to converge uniformly in ℏ ∈ [ 0 , 1 ] . This yields non-trivial examples of quantum integrable systems, an exact quantization formula for the… Expand
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References

SHOWING 1-10 OF 19 REFERENCES
Exercises in exact quantization
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrodinger Hamiltonians [-d2/dq2 + V(q)]± on the half-line {q>0}, with a DirichletExpand
Semi-classical approximation in quantum mechanics
I Quantization of Velocity Field (the Canonical Operator).- 1. The method of Stationary phase. The Legendre Transformation.- 2. Pseudodifferential Operators.- 3. The Hamilton-Jacobi Equation. TheExpand
Normal Forms and Quantization Formulae
Abstract:We consider the Schrödinger operator , where as , is Gevrey of order and has a unique non-degenerate minimum. A quantization formula up to an error of order is obtained for allExpand
Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms I –¶Birkhoff Normal Forms
Abstract. The aim of this paper (part I and II) is to explore the relationship between the effective (Nekhoroshev) stability for near-integrable Hamiltonian systems and the semi-classical asymptoticsExpand
A criterion of integrability for perturbed nonresonant harmonic oscillators. “Wick ordering” of the perturbations in classical mechanics and invariance of the frequency spectrum
We introduce an analogue to the renormalization theory (of quantum fields) into classical mechanics. We also find an integrability criterion guaranteeing the convergence of Birkhoff's series and anExpand
Harmonic analysis in phase space
This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and relatedExpand
Canonical transformations depending on a small parameter
The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter. Lie transforms define naturally a class of canonicalExpand
LIE TRANSFORM PERTURBATION THEORY FOR HAMILTONIAN SYSTEMS
Abstract A review of the theory of Lie transform perturbation theory for Hamiltonian systems is presented. The operator theory of Dewar for continuous families of canonical transformations isExpand
Convergence or generic divergence of the Birkhoff normal form
We prove that the Birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent. The same result is proved for theExpand
A Uniform Quantum Version of the Cherry Theorem
Consider in $$L^2(\mathbb{R}^2)$$ the operator family $$H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$$ . P0 is the quantum harmonic oscillator with diophantine frequency vector ω, F0 a boundedExpand
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