Temperature dependence of collective states in the random-phase approximation

We investigate the temperature dependence of collective states in the framework of the random-phase approximation at finite temperature. We show that sum rules can be extended to collective energies at finite temperature. Numerical methods are developed to solve the RPA equations at finite temperature. Results are presented and discussed in the case of ⁴⁰Ca for isovector dipole and isoscalar octupole vibrations, using oscillator wave functions and a zero-range force. We show that the broadening of giant dipole resonances observed experimentally, appears as a natural consequence of the structure of the RPA equations. Comparison is made with the schematic model for which the temperature dependence of collective states can be worked out analytically.