# A SUPPORT THEOREM FOR A WAVE EQUATION

4-2023

Thesis

## Degree Name

Master of Science in Mathematics

## Department

Mathematical Sciences

Salem Ben Said

## Abstract

It is well known that the fundamental solution to the classical wave equation ฮ๐ข (๐ฅ, ๐ก) โ โ๐ก๐ก๐ข(๐ฅ,๐ก) = 0 is supported on the light cone {(๐ฅ, ๐ก) โ โ๐ร โ : ||๐ฅ|| = |๐ก|} if and only if the dimension ๐ is odd and โฅ 3. Because we are living in a 3-dimensional world we can hear each other clearly; One has a pure propagator without residual waves. In this thesis we consider the wave equation

2||๐ฅ||ฮ๐๐ข๐(๐ฅ, ๐ก) โ โ๐ก๐ก๐ข(๐ฅ,๐ก) = 0, (๐ฅ, ๐ก) โ โ๐ ร โ

where ฮ๐is a second order differential and difference operator. First, we prove the existenceand the uniqueness of the solution ๐ข๐(๐ฅ, ๐ก). Second, we search for the condition on the parameter ๐ and the dimension ๐ for the fundamental solution to be supported on the lightcone {(๐ฅ, ๐ก) โ โ๐ร โ : sqrt(2||๐ฅ||) = |๐ก|}. Our approach is based heavily on the representation theory of the Lie algebra ๐๐ฉ(2, โ), where we construct a new representation ฯ๐ of ๐๐ฉ(2,โ) acting on the Schwartz space ๐บ(โ๐). Finally, we prove that ฯ๐ lifts to give raise to a unitary representation of a simply connected Lie group with Lie algebra ๐๐ฉ(2,โ).

## Arabic Abstract

ูู ุงูุนุฑูู ุฃู ุงูุญู ุงูุฃุณุงุณู ููุนุงุฏูุฉ ุงูููุฌุฉ ุงูููุงุณูููุฉ ฮ๐ข (๐ฅ, ๐ก) โ โ๐ก๐ก๐ข(๐ฅ,๐ก) = 0 ูุชู ุฏุนูู ุนูู ุงููุฎุฑูุท ุงูุถูุฆู {(๐ฅ, ๐ก) โ โ๐ร โ : ||๐ฅ|| = |๐ก|} ุฅุฐุง ูููุท ุฅุฐุง ูุงูุช ุงูุฃุจุนุงุฏ โ ๐ ูุฑุฏูุฉ ูุฃูุจุฑ ุฃู ุชุณุงูู 3. ูุฃููุง ูุนูุด ูู ุนุงูู ุซูุงุซู ุงูุฃุจุนุงุฏุ ูููููุง ุณูุงุน ุจุนุถูุง ุงูุจุนุถ ุจูุถูุญ. ูุจุฐูู ูููู ูุฏููุง ูุงูู ููู ุจุฏูู ููุฌุงุช ุจุงููุฉ. ูู ูุฐู ุงูุฃุทุฑูุญุฉ ูุฏุฑุณ ูุนุงุฏูุฉ ุงูููุฌุฉ:

2||๐ฅ||ฮ๐๐ข๐(๐ฅ, ๐ก) โ โ๐ก๐ก๐ข(๐ฅ,๐ก) = 0, (๐ฅ, ๐ก) โ โ๐ ร โ

ุญูุซ ฮ๐ โูู ุนุงูู ุชูุงุถูู ููุฑูู ูู ุงูุฏุฑุฌุฉ ุงูุซุงููุฉ. ุฃููุฃุ ูุซุจุช ูุฌูุฏ ููุฑุฏูุฉ ุงูุญูโ ๐ข๐(๐ฅ, ๐ก). โุซุงููุงูโ ูุจุญุซ ุนู ุงูุดุฑุท ุนูู ุงููุนุงูู ๐ ูุงูุฃุจุนุงุฏ ๐ ูุฏุนู ุงูุญู ุงูุฃุณุงุณู ุนูู ุงููุฎุฑูุท ุงูุถูุฆูโ {(๐ฅ, ๐ก) โ โ๐ร โ : sqrt(2||๐ฅ||) = |๐ก|}. โูุนุชูุฏ ููุฌูุง ุจุดูู ูุจูุฑ ุนูู ูุธุฑูุฉ ุงูุชูุซูู ูุฌุจุฑ ููโู๐๐ฉ(2,โ) ุญูุซ ูููู ุจุจูุงุก ุชูุซููุง ุฌุฏูุฏุงโ ฯ๐ ูู ๐๐ฉ(2,โ) โูุนูู ุนูู ุงููุถุงุก ุงูุดูุงุฑุฒู ๐บ(โ๐) ุฃุฎูุฑุงุ ูุซุจุช ุฃู ฯ๐ โุชุชูุงูู ูุชุนุทู ุชูุซููุงู ููุญุฏุงู ููุฌููุนุฉ ูู ุจุณูุทุฉ ุฐุงุช ุฌุจุฑ ูู ๐๐ฉ(2,โ).

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