Senior Honors Thesis and Final ARTU Presentation: "Spectral Gap of a Specific Linear Operator

Jiefu Xiang

Abstract: We consider two operators O2 and O3 on a closed subspace of L^2(R^3) that appear in a quantum many-body system in condensed matter physics.  We begin with theorems about linear operators in Hilbert space and deduce that O3(g) behaves like inner product of Tg and g, where T is a linear operator.  Using a change of variable, we obtain an integration form of the linear operator T.  With further simplication, we find eigenvalues and eigenfunctions associated with T, and thus the spectrum of T.  The ultimate goal is to prove a relation between O2 and O3 which is O2(f) + O3(f) >= eps*O2(f) for some eps> 0. 

 

Host: Ron Freiwald