Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

Jacek Zienkiewicz

Studia Mathematica (1997)

  • Volume: 122, Issue: 1, page 15-37
  • ISSN: 0039-3223

Abstract

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Let L be the full laplacian on the Heisenberg group ℍ n of arbitrary dimension n. Then for f ∈ L 2 ( ℍ n ) such that ( I - L ) s / 2 f ∈ L 2 ( ℍ n ) , s > 3/4, for a ϕ ∈ C c ( ℍ n ) we have ʃ ℍ n | ϕ ( x ) | s u p 0 < t ≤ 1 | e ( √ - 1 ) t L f ( x ) | 2 d x ≤ C ϕ ∥ f ∥ W s 2 . On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group ℍ n , then for every s < 1 there exists a sequence f n ∈ L 2 ( ℍ n ) and C n > 0 such that ( I - L ) s / 2 f n ∈ L 2 ( ℍ n ) and for a ϕ ∈ C c ( ℍ n ) we have ʃ ℍ n | ϕ ( x ) | s u p 0 < t ≤ 1 | e ( √ - 1 ) t Δ f n ( x ) | 2 d x ≥ C n ∥ f n ∥ W s 2 , l i m n → ∞ C n = + ∞ .

How to cite

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Zienkiewicz, Jacek. "Initial value problem for the time dependent Schrödinger equation on the Heisenberg group." Studia Mathematica 122.1 (1997): 15-37. <http://eudml.org/doc/216357>.

@article{Zienkiewicz1997,
abstract = {Let L be the full laplacian on the Heisenberg group $ℍ^\{n\}$ of arbitrary dimension n. Then for $f ∈ L^\{2\}(ℍ^\{n\})$ such that $(I-L)^\{s/2\}f ∈ L^\{2\}(ℍ^\{n\})$, s > 3/4, for a $ϕ ∈ C_\{c\}(ℍ^\{n\})$ we have $ʃ_\{ℍ^\{n\}\} |ϕ(x)| sup_\{0 < t≤1\} |e^\{(√-1)tL\}f(x)|^\{2\} dx ≤ C_\{ϕ\} ∥f∥_\{W^\{s\}\}^\{2\}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^\{n\}$, then for every s < 1 there exists a sequence $f_\{n\} ∈ L^\{2\}(ℍ^\{n\})$ and $C_\{n\} > 0$ such that $(I-L)^\{s/2\} f_\{n\} ∈ L^\{2\}(ℍ^\{n\})$ and for a $ϕ ∈ C_\{c\}(ℍ^\{n\})$ we have $ʃ_\{ℍ^\{n\}\} |ϕ(x)| sup_\{0 < t≤1\} |e^\{(√-1)tΔ\} f_\{n\}(x)|^\{2\} dx ≥ C_\{n\} ∥f_\{n\}∥_\{W^\{s\}\}^\{2\}, lim_\{n→∞\}C_\{n\} = +∞$.},
author = {Zienkiewicz, Jacek},
journal = {Studia Mathematica},
keywords = {Schrödinger equation; time dependent Hamiltonian; Sobolev spaces},
language = {eng},
number = {1},
pages = {15-37},
title = {Initial value problem for the time dependent Schrödinger equation on the Heisenberg group},
url = {http://eudml.org/doc/216357},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Zienkiewicz, Jacek
TI - Initial value problem for the time dependent Schrödinger equation on the Heisenberg group
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 1
SP - 15
EP - 37
AB - Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f ∈ L^{2}(ℍ^{n})$ such that $(I-L)^{s/2}f ∈ L^{2}(ℍ^{n})$, s > 3/4, for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tL}f(x)|^{2} dx ≤ C_{ϕ} ∥f∥_{W^{s}}^{2}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$, then for every s < 1 there exists a sequence $f_{n} ∈ L^{2}(ℍ^{n})$ and $C_{n} > 0$ such that $(I-L)^{s/2} f_{n} ∈ L^{2}(ℍ^{n})$ and for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tΔ} f_{n}(x)|^{2} dx ≥ C_{n} ∥f_{n}∥_{W^{s}}^{2}, lim_{n→∞}C_{n} = +∞$.
LA - eng
KW - Schrödinger equation; time dependent Hamiltonian; Sobolev spaces
UR - http://eudml.org/doc/216357
ER -

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