Spectral projections for the twisted Laplacian

Herbert Koch, and Fulvio Ricci. "Spectral projections for the twisted Laplacian." Studia Mathematica 180.2 (2007): 103-110. <http://eudml.org/doc/285179>.

@article{HerbertKoch2007,
abstract = {Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = -1/2 ∑_\{j=1\}^\{n\} [(∂_\{x_\{j\}\} + iy_\{j\})² + (∂_\{y_\{j\}\} - ix_\{j\})²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_\{λ\}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $||P_\{λ\}u||_\{L^\{p\}(ℝ^\{d\})\} ≲ λ^\{ϱ(p)\} ||u||_\{L²(ℝ^\{d\})\}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1) ≤ p ≤ ∞.},
author = {Herbert Koch, Fulvio Ricci},
journal = {Studia Mathematica},
keywords = {spectral projection; twisted Laplacian},
language = {eng},
number = {2},
pages = {103-110},
title = {Spectral projections for the twisted Laplacian},
url = {http://eudml.org/doc/285179},
volume = {180},
year = {2007},
}

TY - JOUR
AU - Herbert Koch
AU - Fulvio Ricci
TI - Spectral projections for the twisted Laplacian
JO - Studia Mathematica
PY - 2007
VL - 180
IS - 2
SP - 103
EP - 110
AB - Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = -1/2 ∑_{j=1}^{n} [(∂_{x_{j}} + iy_{j})² + (∂_{y_{j}} - ix_{j})²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $||P_{λ}u||_{L^{p}(ℝ^{d})} ≲ λ^{ϱ(p)} ||u||_{L²(ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1) ≤ p ≤ ∞.
LA - eng
KW - spectral projection; twisted Laplacian
UR - http://eudml.org/doc/285179
ER -