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Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits

Didier Robert, H. Tamura (1989)

Annales de l'institut Fourier

We study the semi-classical asymptotic behavior as ( h → 0 ) of scattering amplitudes for Schrödinger operators - ( 1 / 2 ) h 2 Δ + V . The asymptotic formula is obtained for energies fixed in a non-trapping energy range and also is applied to study the low energy behavior of scattering amplitudes for a certain class of slowly decreasing repulsive potentials without spherical symmetry.

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian

Pedro Ricardo Simão Antunes, Pedro Freitas, James Bernard Kennedy (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of minimising the nth-eigenvalue of the Robin Laplacian in RN. Although for n = 1,2 and a positive boundary parameter α it is known that the minimisers do not depend on α, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α. We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with n1/N, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further...

Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue

Dimitri Mugnai (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant’s Nodal theorem.

Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue

Dimitri Mugnai (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant's Nodal theorem.

Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov, Georgi Popov (1982)

Annales de l'institut Fourier

Let S ( λ ) be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle 𝒪 ⊂ R n , n ≥ 3 with Dirichlet or Neumann boundary conditions on ∂ 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 ) ∼ ∑ j = 0 ∞ c j λ n - j as λ → + ∞ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

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