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# Advances in Theoretical and Mathematical Physics

## Volume 18 (2014)

### Number 6

### A new approach to the $N$-particle problem in QM

Pages: 1287 – 1334

DOI: http://dx.doi.org/10.4310/ATMP.2014.v18.n6.a3

#### Author

#### Abstract

In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered. The aim is to bring a fermionic Hamiltonian for arbitrary numbers $N$ of particles by analytical means into a shape such that modern numerical methods can successfully be applied. For this purpose the Cook-Schroeck Formalism is taken as starting point. This includes the use of the occupation number representation. It is shown that the $N$-particle Hamiltonian is determined in a canonical way by a fictional 2-particle Hamiltonian. A special approximation of this 2-particle operator delivers an approximation of the $N$-particle Hamiltonian, which is the orthogonal sum of finite dimensional operators. A complete classification of the matrices of these operators is given. Finally the method presented here is formulated as a work program for practical applications. The connection with other methods for solving the same problem is discussed.