Sinh-Gordon Boundary TBA and Boundary Liouville Reflection Amplitude

The ground state energy of the sinh-Gordon model defined on the strip is studied using the boundary thermodynamic Bethe ansatz equation. Its ultraviolet (small width of the strip) behavior is compared with the one obtained from the boundary Liouville reflection amplitude. The results are in perfect agreement in the allowable range of the parameters and provide convincing support for both approaches. We also describe how the ultraviolet limit of the effective central charge can exceed one in the parameter range when the Liouville zero mode forms a bound state.