**Authors**:
G.allavotti, C. Marchioro

**Title
On the calculation of an integral**:
* Journal of Mathematical Analysis and
Applications,
44, 661--675, 1973,
*

**Abstract:** The integral over $R^n$
$$I_n(g,\omega)=\int dq_1\ldots dq_n
\exp\{-\frac12\sum_{i\ne j}^{1,n}
\frac{g^2}{(q_i-q_j)^2}-\frac12\sum_{i=1}^n \omega^2 q_i^2\}$$
is exactly computed and shown to be
$$I_n(g,\omega)=I_n(0,\omega) \exp\{-\omega g \frac{n(n-1)}2\}$$
In Sec. 5 we conjecture that the related Hamiltonian mechanical system
is exactly integral and propose the form it would take in action-angle
variables.

* A Follow Up: The conjecture in Sec.5 has been
proved partially (integrability) in "J. Moser,
Three integrable Hamiltonian systems connected with isospectral
deformations Advances in Mathematics, 16:197--220, 1975",
and completed (action-angle variables found) in "J.P. Francoise.
Canonical partition functions of Hamiltonian systems and the
stationary phase formula,
Communications in Mathematical Physics, 117: 37--47, 1988".
*

**Key words: **
Statistical Mechanics, Path Integral, Brwonian Motion, FEynman-Kac Formula

*Giovanni Gallavotti
Fisica, Universita' di Roma 1
P.le Moro 2
00185 Roma, Italia
tel +39-06-4991-4370
fax +39-06-4957697
*

*em: giovanni.gallavotti@roma1.infn.it
http://ipparco.roma1.infn.it
*