Thermal state of a harmonic oscillator with a linearly decreasing mass

By considering an ensemble system composed of a harmonic oscillator with a linearly decreasing mass, which follows a Bose-Einstein distribution, we investigated the thermal state and obtained the partition function and the density operator. The expectation values of the invariant operator and quantum-mechanical energy in the thermal state were evaluated. Although the mean value of the canonical variables depended on the signal plus noise, the signal plus noise did not affect the fluctuations of the canonical variables; consequently, it did no affect the uncertainty relation. We confirmed that a correspondence existed for the time dependency of energy expectation values between the number state and the thermal state.