Solutions of the Klein-Gordon equation in an infinite square-well potential with a moving wall

Introduction. – The non-relativistic system of a onedimensional infinite square well with a massive particle evolving according to the Schrödinger equation is one of the most elementary quantum mechanical systems, and it often serves as an approximation to more complex physical systems. If, however, the potential walls are allowed to move, as originally in the Fermi-Ulam model for the acceleration of cosmic rays [1, 2], the situation is much more complicated: if one does not choose to rely on an adiabatic approximation, then the system is not separable anymore. Results concerning exact solutions exist only sparsely and have attracted a considerable amount of attention. For the special case of a non-relativistic system with a wall moving at constant velocity, such exact solutions have been obtained first in [3], see [4–6] for generalizations. The relativistic moving-wall system on the other hand is a much less common object of study, and there appear interesting subtleties. This article concerns the onedimensional Klein-Gordon (KG) particle in an infinite square well with a boundary which is moving outward at constant velocity ν. We find an infinite set of exact solutions which do not rely on an adiabatic approximation on top of the square-well approximation with definite position and momentum configuration. We thereby generalize the solutions presented in the appendix of [7], which are valid only for a special case and which are stated without specifying any method on how to obtain them. In contrast, we use a transformation to hyperbolic space which provides in a simple and new manner a set of general solutions for the massless as well as for the massive case, while introducing derivatives of first order into the transformed KG equation. Exact solution. – This article concerns the initial/boundary value problem  ∂ ∂t2 Ψ(t, x) = ∂ 2 ∂x2 Ψ(t, x)−m2Ψ(t, x) in F Ψ|∂F = 0 on ∂F Ψ(t0, x) = f(x), (∂tΨ)(t0, x) = g(x). (1)