By integrating the pressure equation at the surface of a self coupled curvilinear boundary, one may obtain asymptotic estimates of energy shifts, which is especially useful in lattice QCD studies of nonrelativistic bound states. Energy shift expressions are found for periodic (antiperiodic) boundary conditions on antipodal points, which require Neumann (Dirichlet) boundary conditions for even parity states and Dirichlet (Neumann) boundary conditions for odd parity states. It is found that averaging over periodic and antiperiodic boundary conditions is an effective way of removing the asymptotic energy shifts from the boundary. Asymptotic energy shifts from boxes with self coupled walls are also considered and shown to be effectively antipodal. The energy shift equations are illustrated by the solution of the bounded harmonic oscillator and hydrogen atoms.
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