Vacuum energy in smooth background fields

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Abstract
We consider the ground-state energy of a scalar field in the background of a general potential which depends on one coordinate. We consider a general expression following from the analytical properties of the one-dimensional scattering matrix. We show that reflections give a positive and bound states a negative contribution to the ground-state energy and we calculate explicitly two simple examples, the square-well potential and a piecewise oscillatory potential. We demonstrate our formulae by an easy rederivation of the mass of the kink.