  We outline the nonlinear transformation in the path integral representation
for partition function of O(N) symmetric oscillator systems bringing their
duality to certain one-dimensional oscillators with unstable potential shapes.
This duality transformation realizes the equivalence between spectra of a
Hermitian and a non-Hermitian Hamiltonians. It is exploited to calculate
exactly the spectra of quantum rotators . The zero-temperature limit is
considered and the dual representation of n-point correlators of coordinate
operators is obtained.
