Failure of the WKB approximation in calculations of soliton free energies

The authors examine the accuracy of the WKB approximation used in previous work to compute soliton free energies at low temperatures; this examination is crucial in establishing the validity of a simple physical phenomenology as an alternative to the transfer-operator method of calculation. For the ${\phi }^4$ and double sine-Gordon potentials a method developed by Goldstein (1929) is adapted which is known to give correct results in the sine-Gordon case. For all three potentials, the WKB approximation is in error by the factor ${\left(\frac{e}{\pi }\right)}^{\frac{1}{2}}$.