Dispersive Estimates for Four Dimensional Schrödinger and Wave Equations with Obstructions at Zero Energy

By M. Burak, Erdo Gan, M. Goldberg, William R. Green. 2013

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We investigate L 1(ℝ4) → L ∞(ℝ4) dispersive estimates for the Schrödinger operator H = − Δ +V when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator F t satisfying ‖F t ‖ L 1→L ∞ < 1/log t for t > 2 such that We also show that the operator F t = 0 if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5204d3514af993b9f6d286ddef16a6afbee2085e

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In Mathematical Physics

1. William R. Green
2. M. Goldberg

Citations in the corpus

• Dispersive Estimates for Higher Dimensional Schrödinger Operators with Threshold Eigenvalues I: The Odd Dimensional Case (2014, methodology): “In even dimensions the expansions are more complicated, involving terms of the form λk(log λ)l, k ≥ −2, see for example [16, 17, 7, 5, 12].”