Semi-classical quantization rules for a periodic orbit of hyperbolic\n type

2016/07/18 by H. Louati, Louati, Hanen, Michel Rouleux +1 · 2 citations
Physics and Astronomy · Mathematics · #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1607.05057

Abstract

Determination of periodic orbits for a Hamiltonian system together with their\nsemi-classical quantization has been a long standing problem. We consider here\nresonances for a h-Pseudo-Differential Operator H(y,hD_y;h) induced by a\nperiodic orbit of hyperbolic type at energy E₀. We generalize the framework\nof [G 'eSj], in the sense that we allow for both hyperbolic and elliptic\neigenvalues of Poincar 'e map, and show that all resonances in\nW=[E₀-\ε₀,E₀+\ε₀]-i]0,h^\δ], 0<\δ<1, are\ngiven by a generalized Bohr-Sommerfeld quantization rule.\n

Citations

• Morse‐type index theory for flows and periodic solutions for Hamiltonian Equations

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• Bohr-Sommerfeld quantization conditions for Schrodinger operator: the Method of Microlocal Wronskian and Gram Matrix
• Bohr-Sommerfeld Quantization Rules for 1-D Semiclassical Pseudo-Differential Operator: the Method of Microlocal Wronskian and Gram Matrix

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