A trace formula for resonances and application to semi-classical Schrödinger operators

Johannes Sjöstrand

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Poisson formulæ for resonances.

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Semiclassical Resonances and trace formulae for non-semi-bounded Hamiltonians

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Sharp bounds for the number of the scattering poles

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Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov, Georgi Popov (1982)

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Let $S (λ)$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle $𝒪 \subset R^{n}$ , $n \geq 3$ with Dirichlet or Neumann boundary conditions on $\partial 𝒪$ . The function $s (λ)$ , called scattering phase, is determined from the equality $e^{- 2 π i s (λ)} = \det S (λ)$ . We show that $s (λ)$ has an asymptotic expansion $s (λ) \sim \sum_{j = 0}^{\infty} c_{j} λ^{n - j}$ as $λ \to + \infty$ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

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