Uniqueness of solutions to schrödingera equations on H-type groups

Salem Ben Saïd; Sundaram Thangavelu; Venku Naidu Dogga

doi:10.1017/S1446788713000311

Uniqueness of solutions to schrödingera equations on H-type groups

Salem Ben Saïd
, Sundaram Thangavelu
, Venku Naidu Dogga

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

This paper deals with the Schrödinger equation i∂_su(z, t; s) - Lu(z, t; s)= 0, where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies |f(z, t)| ≤ q_a(z, t), where q_s is the heat kernel associated to L. If in addition |u(z, t; s₀) ≤ q_β (z, t), for some s₀ ∈ R \ {0}, then we prove that u(z, t; s)= 0 for all s ∈ R whenever αβ < s₀². This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rⁿ⁺¹.

Original language	English
Pages (from-to)	297-314
Number of pages	18
Journal	Journal of the Australian Mathematical Society
Volume	95
Issue number	3
DOIs	https://doi.org/10.1017/S1446788713000311
Publication status	Published - Dec 2013
Externally published	Yes

Keywords

H-type groups
Heat kernel
Schrödinger equation
Spherical harmonics
Sub-Laplacian

ASJC Scopus subject areas

General Mathematics

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10.1017/S1446788713000311

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@article{84d542987dfe44c99bac93a45921eca2,

title = "Uniqueness of solutions to schr{\"o}dingera equations on H-type groups",

abstract = "This paper deals with the Schr{\"o}dinger equation i∂su(z, t; s) - Lu(z, t; s)= 0, where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies |f(z, t)| ≤ qa(z, t), where qs is the heat kernel associated to L. If in addition |u(z, t; s0) ≤ qβ (z, t), for some s0 ∈ R \textbackslash{} \{0\}, then we prove that u(z, t; s)= 0 for all s ∈ R whenever αβ < s02. This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1.",

keywords = "H-type groups, Heat kernel, Schr{\"o}dinger equation, Spherical harmonics, Sub-Laplacian",

author = "Sa{\"i}d, \{Salem Ben\} and Sundaram Thangavelu and Dogga, \{Venku Naidu\}",

note = "Funding Information: The work of the last two authors is supported by a J. C. Bose Fellowship from DST, India. 12 2013 07 08 2013 95 3 297 314 04 06 2011 19 05 2013 Copyright {\textcopyright}2013 Australian Mathematical Publishing Association Inc. 2013 Australian Mathematical Publishing Association Inc. ",

year = "2013",

month = dec,

doi = "10.1017/S1446788713000311",

language = "English",

volume = "95",

pages = "297--314",

journal = "Journal of the Australian Mathematical Society",

issn = "1446-7887",

publisher = "Cambridge University Press",

number = "3",

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TY - JOUR

T1 - Uniqueness of solutions to schrödingera equations on H-type groups

AU - Saïd, Salem Ben

AU - Thangavelu, Sundaram

AU - Dogga, Venku Naidu

N1 - Funding Information: The work of the last two authors is supported by a J. C. Bose Fellowship from DST, India. 12 2013 07 08 2013 95 3 297 314 04 06 2011 19 05 2013 Copyright ©2013 Australian Mathematical Publishing Association Inc. 2013 Australian Mathematical Publishing Association Inc.

PY - 2013/12

Y1 - 2013/12

N2 - This paper deals with the Schrödinger equation i∂su(z, t; s) - Lu(z, t; s)= 0, where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies |f(z, t)| ≤ qa(z, t), where qs is the heat kernel associated to L. If in addition |u(z, t; s0) ≤ qβ (z, t), for some s0 ∈ R \ {0}, then we prove that u(z, t; s)= 0 for all s ∈ R whenever αβ < s02. This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1.

AB - This paper deals with the Schrödinger equation i∂su(z, t; s) - Lu(z, t; s)= 0, where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies |f(z, t)| ≤ qa(z, t), where qs is the heat kernel associated to L. If in addition |u(z, t; s0) ≤ qβ (z, t), for some s0 ∈ R \ {0}, then we prove that u(z, t; s)= 0 for all s ∈ R whenever αβ < s02. This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1.

KW - H-type groups

KW - Heat kernel

KW - Schrödinger equation

KW - Spherical harmonics

KW - Sub-Laplacian

UR - https://www.scopus.com/pages/publications/84893680843

UR - https://www.scopus.com/pages/publications/84893680843#tab=citedBy

U2 - 10.1017/S1446788713000311

DO - 10.1017/S1446788713000311

M3 - Article

AN - SCOPUS:84893680843

SN - 1446-7887

VL - 95

SP - 297

EP - 314

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 3

ER -

Uniqueness of solutions to schrödingera equations on H-type groups

Abstract

Keywords

ASJC Scopus subject areas

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