Reflection factors and a two-parameter family of boundary bound states in the sinh-Gordon model

The investigation of boundary breather states of the sinh-Gordon model restricted to a half-line is revisited. Properties of the classical boundary breathers for the two-parameter family of integrable boundary conditions are reviewed and extended. The energy spectrum of the quantized boundary states is computed, firstly by using a bootstrap technique and, subsequently using a WKB approximation. Requiring that the two descriptions of the spectrum agree with one another allows a determination of the relationship between the boundary parameters, the bulk coupling constant, and the two parameters appearing in the reflection factor describing the scattering of the sinh-Gordon particle from the boundary. These calculations had been performed previously for the case in which the boundary conditions preserve the bulk $Z_2$ symmetry of the model. The significantly more difficult case of general boundary conditions which violate the bulk symmetry is treated in this article. The results clarify the weak-strong coupling duality of the sinh-Gordon model with integrable boundary conditions.