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Physical Review Letters

Further Results for the Many-Body Problem in One Dimension

Bill Sutherland

Published Jan 15, 1968• 319 citations
Publisher

Abstract

The problem of $N$ particles interacting in one dimension by a repulsive delta-function potential is solved, when the wave function $\ensuremath{\psi}$ transforms like any irreducible representation ${R}_{\ensuremath{\psi}}$ of ${S}_{N}$ for which the Young tableau consists of a finite number of either rows or columns.

DOI
10.1103/physrevlett.20.98

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The problem of NNN particles interacting in one dimension by a repulsive delta-function potential is solved, when the wave function \ensuremathψ\ensuremath{\psi}\ensuremathψ transforms like any irreducible representation R\ensuremathψ{R}_{\ensuremath{\psi}}R\ensuremathψ​ of SN{S}_{N}SN​ for which the Young tableau consists of a finite number of either rows or columns.
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.
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 integral equations for the calculation of the ground-state energy of the repulsive delta-function interaction problem in one dimension for NNN particles are reduced to a system of algebraic equations by using the method of moments. We obtain explicit results in a limiting case.
The problem of one-dimensional particles interacting via a delta-function potential has been a useful model for the many-body problem. While it can be solved for bosons, it has so far not been solved for fermions except for the special cases of either all spins parallel or all but one spin parallel. The full difficulty of the problem first manifests itself when two spins are down (i.e., $S=\frac{1}{2}N\ensuremath{-}2$), and this we solve here. We are confident that our method can be extended to the general problem.
The problem of N bosons interacting with each other via repulsive delta-function interactions and with an external, attractive, delta-function potential is solved, within the Hartree approximation, exactly. If the interparticle interactions are above a certain value, there is no bound state. Thus the bound state does not just expand to compensate for the increase in the repulsive Hartree potential. Moreover as the interaction strength is increased to that value, the ground-state wavefunction develops a pole at the position of the attractive potential.