We compute the boundary energy and the Casimir energy for both the spin-1/2 XXZ quantum spin chain and (by means of the light-cone lattice construction) the massive sine-Gordon model with both left and right boundaries. We also derive a nonlinear integral equation for the ground state of the sine-Gordon model on a finite interval. These results, which are based on a recently-proposed Bethe ansatz solution, are for general values of the bulk coupling constant, and for both diagonal and nondiagonal boundary interactions. However, the boundary parameters are restricted to obey one complex (two real) constraints.
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We are grateful to F. Ravanini for providing us with sample code for numerical solution of the nonlinear integral equation; and to O. Alvarez for his help in preparing a figure. This work was supported in part by the Korea Research Foundation 2002-070-C00025 (C.A.); and by the National Science Foundation under Grants PHY-0098088 and PHY-0244261, and by a UM Provost Award (R.N.).