2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/206666We 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.12 pages, substantial changes: calculation of correlators addedHigh Energy Physics - TheoryMathematical PhysicsNuclear TheoryAtomic PhysicsQuantum Physicstext