Recent results on the separatrix splitting for the standard map

V. F. Lazutkin

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Semiclassical eigenstates in a multidimensional well

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Diffusion times and stability exponents for nearly integrable analytic systems

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For a positive integer n and R>0, we set $B_{R}^{n} = \{x \in ℝ^{n} | {∥x∥}_{\infty} < R\}$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $h (r) = \frac{1}{2} r_{1}^{2} + . . . \frac{1}{2} r_{n - 1}^{2} + r_{n}$ on $𝕋^{n} \times B_{R}^{n}$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $𝕋^{n} \times B_{R}^{n}$ , and setting $ε_{j} : = {∥h - H_{j}∥}_{C^{0} (V)}$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly...