# Quantum and Classical Fidelity for Singular Perturbations of the Inverted and Harmonic Oscillator

Abstract : Let us consider the quantum/versus classical dynamics for Hamiltonians of the form \beq \label{0.1} H_{g}^{\epsilon} := \frac{P^2}{2}+ \epsilon \frac{Q^2}{2}+ \frac{g^2}{Q^2} \edq where $\epsilon = \pm 1$, $g$ is a real constant. We shall in particular study the Quantum Fidelity between $H_{g}^{\epsilon}$ and $H_{0}^{\epsilon}$ defined as \beq \label{0.2} F_{Q}^{\epsilon}(t,g):= \langle \exp(-it H_{0}^{\epsilon})\psi, exp(-itH_{g}^ {\epsilon})\psi \rangle \edq for some reference state $\psi$ in the domain of the relevant operators. We shall also propose a definition of the Classical Fidelity, already present in the literature (\cite{becave1}, \cite{becave2}, \cite{ec}, \cite{prozni}, \cite{vepro}) and compare it with the behaviour of the Quantum Fidelity, as time evolves, and as the coupling constant $g$ is varied.
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Journal articles

Cited literature [14 references]

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Contributor : Sylvie Flores <>
Submitted on : Wednesday, March 15, 2006 - 9:49:29 AM
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### Citation

M. Combescure, A. Combescure. Quantum and Classical Fidelity for Singular Perturbations of the Inverted and Harmonic Oscillator. Journal of Mathematical Analysis and Applications, Elsevier, 2006, 326, pp.908-928. ⟨10.1016/j.jmaa.2006.03.044⟩. ⟨in2p3-00025779⟩

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