We derive a nonlinear integral equation (NLIE) for some bulk excited states of the sine-Gordon model on a finite interval with general integrable boundary interactions, including boundary terms proportional to the first time derivative of the field. We use this NLIE to compute numerically the dimensions of these states as a function of scale, and check the UV and IR limits analytically. We also find further support for the ground-state NLIE by comparison with boundary conformal perturbation theory (BCPT), boundary truncated conformal space approach (BTCSA) and the boundary analogue of the Lüscher formula.
Bibliographical noteFunding Information:
We are grateful to F. Ravanini for valuable discussions. This work was supported in part by the Korea Research Foundation 2002-070-C00025 (C.A.); by the EC network “EUCLID”, contract number HPRN-CT-2002-00325, and Hungarian research funds OTKA D42209, T037674, T034299 and T043582 (Z.B., L.P. and G.T.); and by the National Science Foundation under Grants PHY-0098088 and PHY-0244261, and by a UM Provost Award (R.N.).