One-loop analysis with nonlocal boundary conditions

In the 1980s, Schröder studied a quantum mechanical model where the stationary states of Schrödinger’s equation obey nonlocal boundary conditions on a circle in the plane. For such a problem, we perform a detailed one-loop calculation for three choices of the kernel characterizing the nonlocal boundary conditions. In such cases, the ζ(0) value is found to coincide with the one resulting from Robin boundary conditions. The detailed technique here developed may be useful for studying one-loop properties of quantum field theory and quantum gravity if nonlocal boundary conditions are imposed.