Excitation Spectrum of a One-Dimensional Many-Boson System

The lowest order correction term to the Bogoliubov spectrum IS calculated analytically for the case of one-dimensional bosons with a-function interaction. We obtain the spectrum E(p) = IPI vp'+4,;c + (2nchr)r 112 ·[p[/vp'+4,~c+O(rnc), (n=density, h=2m=1, strength of o-function=2c, r=c/n). This is compared with Lieb's exact excitation spectrum for this case. Our numerical calculation shows that it is the correct r'12 expansion of the spectrum. However this expansion up to second order is not a good approximation for the low wave number part and gives an incorrect expansion of the sound velocity up to second order. 33 We consider the excitation spectrum of many-boson system interacting with two-body potential. The Hamiltonian is (1) BogoliubovlJ derived an approximate excitation spectrum of this system 2 ( p) = [p f X J jl+2nv-(p), where n is the density of particles.*) Recently many authors con-sidered the lowest order correction to this Bogoliubov spectrum. Lee2) derived it from Bogoliubov and Zubarev3> (BZ) Hamil toni an and from correlated basis function (CBF) method in the uniform limit!) He succeeded in showing that both the results are equivalent. Tsuzuki, 5l Grest and Rajagopal6) calculated it from Suna-kawa, Yamasaki and Kebukawa (SYK) Hamiltonian.7l Rajagopal and Grest8) showed that the result of SYK is equivalent to that of BZ. More recently Grest and Rajagopal9> derived it from Bogoliubov's canonical transformation method (B) and showed that the result is equivalent to that of BZ. In these theories zeroth order spectrum i