  We introduce a one-parameter deformation for one-dimensional (1D) quantum
lattice models, the hyperbolic deformation, where the scale of the local energy
is proportional to cosh lambda j at the j-th site. Corresponding to a 2D
classical system, the deformation does not strongly modify the ground state. In
this situation, the effective Hamiltonian of the quantum system shows that the
quasi particle is weakly bounded around the center of the system. By analyzing
this binding effect, we derive scaling relations for the mean-square width
<w^2> of confinement, the energy correction with respect to the excitation gap
\Delta, and the deformation parameter $\lambda$. This finite-size scaling
allows us to investigate excitation gap of 1D non-deformed bulk quantum
systems.
