B\"acklund Transformation for the BC-Type Toda Lattice

We study an integrable case of n-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a B"acklund transformation and prove its basic properties: canonicity, commutativity and spectrality. The B"acklund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n+2, which are mutually dual, sharing the same spectral curve.