Semiclassical energy levels of sine-Gordon model on a strip with Dirichlet boundary conditions

We derive analytic expressions of the semiclassical energy levels of sine-Gordon model in a strip geometry with Dirichlet boundary condition at both edges. They are obtained by initially selecting the classical backgrounds relative to the vacuum or to the kink sectors, and then solving the Schrödinger equations (of Lamè type) associated to the stability condition. Explicit formulas are presented for the classical solutions of both the vacuum and kink states and for the energy levels at arbitrary values of the size of the system. Their ultraviolet and infrared limits are also discussed.