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Effondrements et petites valeurs propres des formes différentielles

Pierre Jammes (2004/2005)

Séminaire de théorie spectrale et géométrie

À courbure et diamètre bornés, les valeurs propres non nulles du laplacien de Hodge-de Rham agissant sur les formes différentielles d’une variété compacte ne sont pas uniformément minorées comme c’est le cas pour les fonctions, et si l’une d’elle tend vers zéro alors le volume de la variété tend aussi vers zéro, c’est-à-dire qu’elle s’effondre. On présente ici les résultats obtenus ces dernières années concernant le problème réciproque, à savoir déterminer le comportement asymptotique des premières...

Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Rivière (2014)

Annales de l’institut Fourier

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of β -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of β -damped trajectories of the geodesic flow.The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues...

Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps

Victor Ivrii (1998/1999)

Séminaire Équations aux dérivées partielles

Asymptotics with sharp remainder estimates are recovered for number N ( τ ) of eigenvalues of the generalized Maxwell problem and for related Laplacians which are similar to Neumann Laplacian. We consider domains with ultra-thin cusps (with exp ( - | x | m + 1 ) width ; m > 0 ) and recover eigenvalue asymptotics with sharp remainder estimates.

Eigenvalues of the p -Laplacian in 𝐑 N with indefinite weight

Yin Xi Huang (1995)

Commentationes Mathematicae Universitatis Carolinae

We consider the nonlinear eigenvalue problem - div ( | ∇ u | p - 2 ∇ u ) = λ g ( x ) | u | p - 2 u in 𝐑 N with p > 1 . A condition on indefinite weight function g is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in W 1 , p ( 𝐑 N ) . A nonexistence result is also given for the case p ≥ N .

Eight exactly solvable complex potentials in Bender-Boettcher quantum mechanics

Znojil, Miloslav (2001)

Proceedings of the 20th Winter School "Geometry and Physics"

This is a readable review of recent work on non-Hermitian bound state problems with complex potentials. A particular example is the generalization of the harmonic oscillator with the potentials: V ( x ) = ω 2 2 x - 2 i β ω 2 - ω 2 . Other examples include complex generalizations of the Morse potential, the spiked radial harmonic potential, the Kratzer-Coulomb potential, the Rosen Morse oscillator and others. Instead of demanding Hermiticity H = H * the condition required is H = P T H P T where P changes the parity and T transforms i to - i .

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