  Specific heats of quantum systems with symmetric and asymmetric double-well
potentials have been calculated. In numerical calculations of their specific
heats, we have adopted the combined method which takes into account not only
eigenvalues of $\epsilon_n$ for $0 \leq n \leq N_m$ obtained by the
energy-matrix diagonalization but also their extrapolated ones for $N_m+1 \leq
n < \infty$ ($N_m=20$ or 30). Calculated specific heats are shown to be rather
different from counterparts of a harmonic oscillator. In particular, specific
heats of symmetric double-well systems at very low temperatures have the
Schottky-type anomaly, which is rooted to a small energy gap in low-lying
two-level eigenstates induced by a tunneling through the potential barrier. The
Schottky-type anomaly is removed when an asymmetry is introduced into the
double-well potential. It has been pointed out that the specific-heat
calculation of a double-well system reported by Feranchuk, Ulyanenkov and
Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order
operator method they adopted neglects crucially important off-diagonal
contributions.
