Applications in physics of the multiplicative anomaly formula involving some basic differential operators

In the framework leading to the multiplicative anomaly formula - which is here proven to be valid even in cases of known spectrum but non-compact manifold (very important in Physics) - zeta-function regularisation techniques are shown to be extremely efficient. Dirac-like operators and harmonic oscillators are investigated in detail, in any number of space dimensions. They yield a non-zero anomaly which, on the other hand, can always be expressed by means of a simple analytical formula. These results are used in several physical examples, where the determinant of a product of differential operators is not equal to the product of the corresponding functional determinants. The simplicity of the Hamiltonian operators chosen is aimed at showing that such a situation may be quite widespread in mathematical physics. However, the consequences of the existence of the determinant anomaly have often been overlooked. (C) 1998 Elsevier Science B.V.