Spectral decomposition and Bloch equation of the operators represented by fixed-centroid path integrals

Interesting approaches to study statical and dynamic properties of quantum systems, e.g., the quantum transition state theory and the centroid molecular dynamics, have been previously derived using fixed centroid path integrals. We show that these constrained propagators can be alternatively defined using an operator formalism. An interesting result is the finding of the differential equations that determine the temperature dependence of these propagators. One equation applies to path integrals with fixed-centroid position (i.e., those used in quantum transition state theory), and the other one to path integrals with fixed-centroid position and momentum (i.e., those used in centroid molecular dynamics). Both equations are solved for a harmonic oscillator, so that the spectral decomposition of the operators represented by fixed-centroid path integrals is derived for this particular case. Their eigenvalues build an alternating geometric series, showing explicitly the impossibility of considering such operat...