Asymptotic Analysis - Volume 103, issue 4

Authors: Chabu, Victor

Article Type: Research Article

Abstract: In this article we study the propagation of Wigner measures linked to solutions of the Schrödinger equation with potentials presenting conical singularities and show that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. We describe in detail either the flow and the mass concentration around and on the singular set and illustrate with examples some issues raised by the …lack of unicity for the classical trajectories at the singularities despite the unicity of the quantum solutions, dismissing any classical selection principle, but in some cases being able to fully solve the propagation problem. Show more

Keywords: Schrödinger equation, Wigner measures, two-microlocal measures, symbolic calculus, quantum-classical correspondence, wave packet approximation

DOI: 10.3233/ASY-171423

Citation: Asymptotic Analysis, vol. 103, no. 4, pp. 165-220, 2017

Authors: Langenau, Holger

Article Type: Research Article

Abstract: The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L 2 on the interval ( − 1 , 1 ) with the weight ( 1 − t 2 ) α and ( 1 − t 2 ) β , respectively. Under the assumption that β − α …is larger than the order of the derivative, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity. Show more

Keywords: Markov inequality, Gegenbauer weight, matrix norm

DOI: 10.3233/ASY-171425

Citation: Asymptotic Analysis, vol. 103, no. 4, pp. 221-233, 2017

Article Type: Other

Citation: Asymptotic Analysis, vol. 103, no. 4, pp. 235-235, 2017