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We study the scaling limit of a statistical system, which is a special caseof the integrable inhomogeneous six-vertex model. It possesses$U_q\big(\mathfrak{sl}(2)\big)$ invariance due to the choice of open boundaryconditions imposed. An interesting feature of the lattice theory is that thespectrum of scaling dimensions contains a continuous component. By applying theODE/IQFT correspondence and the method of the Baxter $Q$ operator thecorresponding density of states is obtained. In addition, the partitionfunction appearing in the scaling limit of the lattice model is computed, whichmay be of interest for the study of nonrational CFTs in the presence ofboundaries. As a side result of the research, a simple formula for the matrixelements of the $Q$ operator for the general, integrable, inhomogeneoussix-vertex model was discovered, that has not yet appeared in the literature.It is valid for a certain one parameter family of diagonal open boundaryconditions in the sector with the $z\,$-projection of the total spin operatorbeing equal to zero.

Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ invariant boundary conditions