Mathematical Physics and Quantum Field Theory
Electron. J. Diff. Eqns., Conf. 04, 2000, pp. 87-101.

Spectral Riesz--Cesàro means: How the square root function helps us to see around the world

S. A. Fulling, E. V. Gorbar, & C. T. Romero

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.

Published July 12, 2000.
Mathematics Subject Classifications: 35P20, 40G05, 81Q10.
Key words: Riesz means, spectral asymptotics, heat kernel, line bundle.

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S. A. Fulling
Department of Mathematics
Texas A&M University
College Station, Texas 77843-3368
E. V. Gorbar
Instituto di Fisica Teorica
Rua Pamplona, 145
0145-900, São Paulo, Brazil
  C. T. Romero
Department of Computer Science
Texas A&M University
College Station, Texas 77843-3112

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