Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Trace expansions for elliptic cone operators with stationary domains


Authors: Juan B. Gil, Thomas Krainer and Gerardo A. Mendoza
Journal: Trans. Amer. Math. Soc. 362 (2010), 6495-6522
MSC (2010): Primary 58J35; Secondary 35P05, 47A10
DOI: https://doi.org/10.1090/S0002-9947-2010-05283-3
Published electronically: July 20, 2010
MathSciNet review: 2678984
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze the behavior of the trace of the resolvent of an elliptic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal extension of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first component and expand the component of finite rank in the case where the domain is stationary. The results make use of, and develop further, our previous investigations on the analytic and geometric structure of the resolvent. The analysis of nonstationary domains, considerably more intricate, is pursued elsewhere.


References [Enhancements On Off] (What's this?)

  • 1. J. Brüning and R. Seeley, Regular singular asymptotics, Adv. in Math. 58 (1985), 133-148. MR 814748 (87b:41032)
  • 2. -, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (2) (1987) 369-429. MR 899656 (88g:35151)
  • 3. -, The expansion of the resolvent near a singular stratum of conical type, J. Funct. Anal. 95 (1991), no. 2, 255-290. MR 1092127 (93g:58146)
  • 4. C. Callias, The heat equation with singular coefficients I. Operators of the form $ -d^{2}/dx^{2}+k/x^{2}$ in dimension $ 1$, Comm. Math. Phys. 88 (3) (1983) 357-385. MR 701923 (84m:58136)
  • 5. J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. USA 76 (1979), 2103-2106. MR 530173 (80k:58098)
  • 6. H. Falomir, M.A. Muschietti, P.A.G. Pisani, On the resolvent and spectral functions of a second order differential operator with a regular singularity, J. Math. Phys. 45 (12) (2004) 4560-4577. MR 2105208 (2005m:81093)
  • 7. H. Falomir, M.A. Muschietti, P.A.G. Pisani, and R. Seeley, Unusual poles of the $ \zeta$-functions for some regular singular differential operators, J. Phys. A 36 (2003), no. 39, 9991-10010. MR 2024508 (2004k:58049)
  • 8. H. Falomir, P.A.G. Pisani, and A. Wipf, Pole structure of the Hamiltonian $ \zeta$-function for a singular potential, J. Phys. A 35 (2002), 5427-5444. MR 1916056 (2003g:81054)
  • 9. J. Gil, Heat trace asymptotics for cone differential operators, Ph.D. thesis, Universität Potsdam, 1998.
  • 10. -, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25-57. MR 1956600 (2004c:58053)
  • 11. J. Gil, T. Krainer, and G. Mendoza, Geometry and spectra of closed extensions of elliptic cone operators, Canad. J. Math. 59 (2007), no. 4, 742-794. MR 2338233 (2008f:58030)
  • 12. -, Resolvents of elliptic cone operators, J. Funct. Anal. 241 (2006), no. 1, 1-55. MR 2264246 (2007j:58032)
  • 13. -, On rays of minimal growth for elliptic cone operators, Oper. Theory Adv. Appl. 172 (2007), 33-50. MR 2308502 (2008c:58022)
  • 14. -, Dynamics on Grassmannians and resolvents of cone operators, to appear in Analysis & PDE.
  • 15. J. Gil and P. Loya, Resolvents of cone pseudodifferential operators, asymptotic expansions and applications, Math. Z. 259 (2008), no. 1, 65-95. MR 2375616 (2009a:58029)
  • 16. J. Gil and G. Mendoza, Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003), no. 2, 357-408. MR 1963689 (2004h:58032)
  • 17. P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, CRC Press, Boca Raton, FL, 1995, second edition. MR 1396308 (98b:58156)
  • 18. G. Grubb and R. Seeley, Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Invent. Math. 121, 481-529 (1995). MR 1353307 (96k:58216)
  • 19. K. Kirsten, P. Loya, and J. Park, The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $ -d\sp 2/dr\sp 2-1/(4r\sp 2)$, J. Math. Phys. 47 (2006), no. 4, 043506, 27 pp. MR 2226343 (2007c:58050)
  • 20. -, Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone, Manuscripta Math. 125 (2008), no. 1, 95-126. MR 2357751 (2009h:58073)
  • 21. -, Exotic expansions and pathological properties of $ \zeta$-functions on conic manifolds, J. Geom. Anal. 18 (2008), no. 3, 835-888. MR 2420767 (2009j:58051)
  • 22. M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Math. vol 136, B.G. Teubner, Stuttgart, Leipzig, 1997.
  • 23. P. Loya, On the resolvent of differential operators on conic manifolds, Comm. Anal. Geom. 10 (2002), no. 5, 877-934. MR 1957656 (2004d:58035)
  • 24. -, Parameter-dependent operators and resolvent expansions on conic manifolds, Illinois J. Math. 46 (2002), no. 4, 1035-1059. MR 1988248 (2005c:58049)
  • 25. P. Loya, P. McDonald, and J. Park, Zeta regularized determinants for conic manifolds, J. Funct. Anal. 242 (2007), no. 1, 195-229. MR 2274020 (2007g:58036)
  • 26. R. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, A K Peters, Ltd., Wellesley, MA, 1993. MR 1348401 (96g:58180)
  • 27. E. Mooers, Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999) 1-36. MR 1714065 (2000g:58039)
  • 28. B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, Studies in Mathematics and its Applications, 24. North-Holland Publishing Co., Amsterdam, 1991. MR 1142574 (93b:47109)
  • 29. R. Seeley, Complex powers of an elliptic operator, Singular Integrals, AMS Proc. Symp. Pure Math. X, 1966, Amer. Math. Soc., Providence, 1967, pp. 288-307. MR 0237943 (38:6220)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58J35, 35P05, 47A10

Retrieve articles in all journals with MSC (2010): 58J35, 35P05, 47A10


Additional Information

Juan B. Gil
Affiliation: Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601-3760

Thomas Krainer
Affiliation: Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601-3760

Gerardo A. Mendoza
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

DOI: https://doi.org/10.1090/S0002-9947-2010-05283-3
Keywords: Resolvents, trace asymptotics, manifolds with conical singularities, spectral theory
Received by editor(s): November 24, 2008
Published electronically: July 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society