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A lower bound for the ground state energy of a Schrödinger operator on a loop

2006
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Proceedings of the American Mathematical Society
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Consider a one-dimensional quantum mechanical particle described by the Schrödinger equation on a closed curve of length 2π. Assume that the potential is given by the square of the curve's curvature. We show that in this case the energy of the particle cannot be lower than 0.6085. We also prove that it is not lower than 1 (the conjectured optimal lower bound) for a certain class of closed curves that have an additional geometrical property.

doi:10.1090/s0002-9939-06-08483-8
fatcat:l6c3ik4lb5a4belf3whgwylm6a