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 Zz symmetry of the model. The significantly more difficult case of general boundary conditions which violate the bulk symmetry is treated in this paper. The results clarify the weak-strong-coupling duality of the sinh-Gordon model with integrable boundary conditions.