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A Dispersion Relation for the Density of States With Application to the Casimir Effect

The trace of a function of a Schrodinger operator minus the same for the Laplacian can be expressed in terms of the determinant of its scattering matrix. The naive formula for this determinant is divergent. Using a dispersion relation, we find another expression for it which is convergent, but needs one piece of information beyond the scattering matrix. Except for this `anomaly', we can express the Casimir energy of a compact body in terms of its optical scattering matrix, without assuming any rotational symmetry for its shape.

preprint2010arXivOpen access
S. G. Rajeev
math-phmath.MPhep-thquant-ph
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S. G. Rajeev

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