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American Journal of Physics

A class of exactly soluble many-body Hamiltonians

Antonio Redondo

Published Jul 1, 1986• 2 citations
Publisher

Abstract

A class of exactly soluble N-particle Hamiltonians is presented. The matrix representation of these Hamiltonians, which consists of one- and two-body interactions, is block diagonal. The blocks consist of a finite-dimensional matrices which can be easily diagonalized. An example for a system with two particles is discussed.
DOI
10.1119/1.14523

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An exactly soluble one-dimensional three-body problem is presented, in which the interaction between the particles consists of local two-body potentials between each two particles. Infinitely high step functions are chosen for the form of the three potential functions. This interaction allows only three-body bound states and no continuum states. We have considered three different choices of the mass ratios of the three particles and we give formulas in closed form for the energies and for the wavefunctions of all states.
In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.
The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.
For both identical and distinguishable particles and for an arbitrary polynomial approximation to the realistic two-body forces, the A-body Hamiltonian may be converted into an infinite block-three-diagonal matrix in the properly arranged, translationally invariant, oscillator basis. As a consequence, the exact Green function and all the projections of eigenstates of the microscopic Schrodinger equation become expressible in terms of the matrix continued fractions. This generalises the recent reformulation of the one-dimensional A=1 anharmonic problem by Graffi and Greechi (1975). The quick convergence of this non-perturbative method of solving the many-body bound-state problem is demonstrated for the simplest three- and four-body examples. With the core-possessing type V(r)=-r 2 +r 4 of the spin- and isospin-independent two-body interaction, only the 10 -3 error in the three-bosonic ground-state energy arises from restricting the formulae to 5*5 dimensional matrices.