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Progress of Theoretical Physics Supplement

Complex Spectral Representation of a Many-Body System in One Dimension

Gonzalo E. Ordonez

Published Jan 1, 2010• 2 citations
Publisher

Abstract

I consider a Brownian particle immersed in a one-dimensional boson gas with delta-function repulsive potentials. I find many-body complex eigenvalues of the Liouville super-operator, up to second order in the coupling constant. The eigenvalues give the diffusion time-scale of the Brownian particle. The eigenvalue problem is solved using a Bethe-ansatz approach.

DOI
10.1143/ptps.184.466

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I consider a Brownian particle immersed in a one-dimensional boson gas with delta-function repulsive potentials. I find many-body complex eigenvalues of the Liouville super-operator, up to second order in the coupling constant. The eigenvalues give the diffusion time-scale of the Brownian particle. The eigenvalue problem is solved using a Bethe-ansatz approach.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.
Making use of a simple unitary transformation we change the Hamiltonian of a particle coupled to a one-dimensional gas of bosons or fermions to a new form from which the many-body degrees of freedom can be easily eliminated by taking the trace. The effective dynamics of the particle allows us to compute its damping constant in the semiclassical approximation in terms of the reflection coefficient of the interaction potential and the occupation number of the environmental particles. We apply our results to a delta-function repulsive potential.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.