We explore a variational approach to the finite-volume $N$ -body problem. The general formalism for $N$ nonrelativistic spinless particles interacting with periodic pairwise potentials yields $N$ -body secular equations. The solutions depend on the infinite-volume $N$ -body wave functions. Given that the infinite-volume $N$ -body dynamics may be solved by the standard Faddeev approach, the variational $N$ -body formalism can provide a convenient numerical framework for finding discrete energy spectra in periodic lattice structures.We explore a variational approach to the finite-volume $N$-body problem. The general formalism for $N$ nonrelativistic spinless particles interacting with periodic pairwise potentials yields $N$-body secular equations. The solutions depend on the infinite-volume $N$-body wave functions. Given that the infinite-volume $N$-body dynamics may be solved by the standard Faddeev approach, the variational $N$-body formalism can provide a convenient numerical framework for finding discrete energy spectra in periodic lattice structures.

arXiv:1810.01261Physical review. D/Physical review. D.Nov 8, 201856 citations

3 Asymptotic solutions of the many-body problem

Виктор Павлович Маслов, O. Yu. Shvedov

De Gruyter eBooksJun 7, 20220 citations

Equilibrium states of a Bose gas with repulsive interactions

Ola Bratteli, Derek W. Robinson

Abstract We consider infinite volume limit Gibbs states of a nonrelativistic quantum Bose gas consisting of one species of spinless particles with positive interaction potentials. The finite volume reduced density matrices are dominated by the corresponding matrices for the noninteracting gas, and as a consequence all infinite volume limit states are regular, locally normal, and analytic on the appropriate CCR algebra. For sufficiently short range repulsive two-body interactions, the cyclic vector associated with the limit state is separating for the σ-weak closure of the algebra in the associated representation.Abstract We consider infinite volume limit Gibbs states of a nonrelativistic quantum Bose gas consisting of one species of spinless particles with positive interaction potentials. The finite volume reduced density matrices are dominated by the corresponding matrices for the noninteracting gas, and as a consequence all infinite volume limit states are regular, locally normal, and analytic on the appropriate CCR algebra. For sufficiently short range repulsive two-body interactions, the cyclic vector associated with the limit state is separating for the σ-weak closure of the algebra in the associated representation.

ANZIAM JournalOct 1, 19809 citations

Many-Body Problem in Quantum Statistical Mechanics. I. General Formulation

T. D. Lee, Chongming Yang

A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain $U$ functions defined for the same system with Boltzmann statistics. It is then shown that these $U$ functions can be evaluated in successive approximations in terms of a binary kernel $B$ which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain $U$ functions defined for the same system with Boltzmann statistics. It is then shown that these $U$ functions can be evaluated in successive approximations in terms of a binary kernel $B$ which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.

Physical ReviewMar 1, 1959210 citations

We explore a variational approach to the finite-volume

N

-body problem. The general formalism for

N

nonrelativistic spinless particles interacting with periodic pairwise potentials yields

N

-body secular equations. The solutions depend on the infinite-volume

N

-body wave functions. Given that the infinite-volume

N

-body dynamics may be solved by the standard Faddeev approach, the variational

N

-body formalism can provide a convenient numerical framework for finding discrete energy spectra in periodic lattice structures.

A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain

U

functions defined for the same system with Boltzmann statistics. It is then shown that these

U

functions can be evaluated in successive approximations in terms of a binary kernel

B

which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.