A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

2005/08/11 by Michael Goldberg, M. Goldberg, Monica Visan +1 · 35 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Class (philosophy) #Counterexample #Differentiable function #Dimension (graph theory) #Interval (graph theory) #Nonlinear Partial Differential Equations #Pointwise #Scaling #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP

paper · pdf · doi:10.1007/s00220-006-0013-5

published in Communications in Mathematical Physics 266(1), 211-238 (Springer Science+Business Media)

arxiv created 2005/08/11 · openalex publication_date 2006/04/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In dimension n>3 we show the existence of a compactly supported potential in the differentiability class C^α, α< \fracn-32, for which the solutions to the linear Schrödinger equation in \Rⁿ, -i∂_t u = - Δu + Vu, u(0)=f, do not obey the usual L¹→ L^∞ dispersive estimate. This contrasts with known results in dimensions n ≤ 3, where a pointwise decay condition on V is generally sufficient to imply dispersive bounds.

Citations

• Spectral properties of Schrödinger operators and time-decay of the wave functions
• Time decay for solutions of Schr�dinger equations with rough and time-dependent potentials
• Decay estimates for Schrödinger operators
• Dispersive Estimates for Schr�dinger Operators in Dimensions One and Three
• Schrödinger semigroups