1 4 Fe b 20 09 The semi - classical spectrum and the Birkhoff normal form
نویسنده
چکیده
• To propose a direct and “elementary” proof of the main result of [3], namely that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the whole semi-classical Birkhoff normal form (denoted the BNF) in the non-resonant case. I believe however that the method used in [3] (trace formulas) are more general and can be applied to any non degenerate non resonant critical point provided that the corresponding critical value is “simple”.
منابع مشابه
er si on 1 - 1 2 Fe b 20 08 A semi - classical inverse problem I : Taylor expansions . ( to Hans Duistermaat for his 65 birthday )
In dimension 1, we show that the Taylor expansion of a “generic” potential near a non degenerate critical point can be recovered from the knowledge of the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Contrary to the work of previous authors, we do not assume that the potential is even. The classical Birkhoff normal form does not contain e...
متن کاملThe semi-classical spectrum and the Birkhoff normal form
• To propose a direct and “elementary” proof of the main result of [3], nameley that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the whole semi-classical Birkhoff normal form (denoted the BNF) in the non-resonant case. I believe however that the method used in [3] (trace formulas) are more general and can be applied to any non degenerate non resonan...
متن کاملBirkhoff Normal Forms in Semi-classical Inverse Problems
The purpose of this note is to apply the recent results on semi-classical trace formulæ [17], and on quantum Birkhoff normal forms for semi-classical Fourier Operators [12] to inverse problems. We show how the classical Birkhoff normal form can be recovered from semi-classical spectral invariants. In fact the full quantum Birkhoff normal form of the quantum Hamiltonian near a closed orbit, and ...
متن کاملA semi - classical inverse problem I : Taylor expansions . ( to the memory of Hans Duistermaat )
In dimension 1, we show that the Taylor expansion of a “generic” potential near a non degenerate critical point can be recovered from the knowledge of the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Contrary to the work of previous authors, we do not assume that the potential is even. The classical Birkhoff normal form does not contain e...
متن کاملar X iv : m at h / 06 08 61 7 v 1 [ m at h . SP ] 2 4 A ug 2 00 6 “ BOTTOM OF THE WELL ” SEMI - CLASSICAL TRACE INVARIANTS
LetˆH be an-admissible pseudodifferential operator whose principal symbol, H, has a unique non-degenerate global minimum. We give a simple proof that the semi-classical asymptotics of the eigenvalues ofˆH corresponding to the " bottom of the well " determine the Birkhoff normal form of H at the minimum. We treat both the resonant and the non-resonant cases.
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