publication . Preprint . Article . 2013

The exotic heat-trace asymptotics of a regular-singular operator revisited

Boris Vertman;
Open Access English
  • Published: 30 Jan 2013
We discuss the exotic properties of the heat-trace asymptotics for a regular-singular operator with general boundary conditions at the singular end, as observed by Falomir, Muschietti, Pisani and Seeley as well as by Kirsten, Loya and Park. We explain how their results alternatively follow from the general heat kernel construction by Mooers, a natural question that has not been addressed yet, as the latter work did not elaborate explicitly on the singular structure of the heat trace expansion beyond the statement of non-polyhomogeneity of the heat kernel.
free text keywords: Mathematics - Spectral Theory, Mathematical Physics, Statistical and Nonlinear Physics, Resolvent, Manifold, Boundary value problem, Differential operator, Pure mathematics, Gravitational singularity, Operator (computer programming), Mathematical analysis, Heat kernel, Riemann zeta function, symbols.namesake, symbols, Mathematics

[AbSt92] Handbook of mathematical functions with formulas, graphs, and mathematical tables, Edited by M. Abramowitz and I. A. Stegun, Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. xiv+1046 pp. ISBN: 0-486-61272-4 MR 1225604 (94b:00012)

[BrSe87] J. Bru¨ning and R. Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), no. 2, 369-429. MR 899656 (88g:35151)

[FMPS03] H. Falomir, M. A. Muschietti, P. A. G. Pisani, and R. Seeley, Unusual poles of the ζ-functions for some regular singular differential operators, J. Phys. A 36 (2003), no. 39, 9991-10010. MR 2024508 (2004k:58049) [OpenAIRE]

[GKM10] J. B. Gil, T. Krainer, G. A. Mendoza, Trace expansions for elliptic cone operators with stationary domains, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6495-6522. MR 2678984 (2011h:58040)

[KLP06] K. Kirsten, P. Loya, and J. Park, The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2/dr2 − 1/(4r2), J. Math. Phys. 47 (2006), no. 4, 043506, 27. MR 2226343 (2007c:58050)

[KLP08] K. Kirsten, P. Loya, and J. Park, Exotic expansions and pathological properties of ζ-functions on conic manifolds, J. Geom. Anal. 18 (2008), no. 3, 835-888. MR 2420767 (2009j:58051)

[Les97] M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997. MR 1449639 (98d:58174)

[LeVe11] M. Lesch, B. Vertman, Regular singular Sturm-Liouville operators and their zeta-determinants, J. Funct. Anal. 261 (2011), no. 2, 408-450. MR 2793118 (2012g:58064)

[Moo99] E. A. Mooers, Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999), 1-36. MR 1714065 (2000g:58039)

[Ver09] B. Vertman, Zeta determinants for regular-singular Laplace-type operators, J. Math. Phys. 50 (2009), no. 8, 083515, 23 pp. MR 2554443 (2010i:58035) [OpenAIRE]

Mathematisches Institut, Universita¨t Bonn, 53115 Bonn, Germany

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