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Spectral Riesz-Cesàro means: How the square root function helps us to see around the world. (English) Zbl 0962.35134
Summary: The heat-kernel expansion for a nonanalytic function of a differential operator, and the integrated (Cesàro-smoothed) spectral densities associated with the corresponding nonanalytic function of the spectral parameter, exhibit a certain nonlocal behavior. Because of this phenomenon, care is needed in applying the pseudodifferential symbolic calculus to nonanalytic functions. We demonstrate this effect by both analytical and numerical calculations for the square root of the Laplace operator in the model of a “twisted” scalar field over the circle. This shows that the effect cannot be attributed solely to boundaries.
MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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