Trace expansions for elliptic cone operators with stationary domains

Gil, Juan; Krainer, Thomas; Mendoza, Gerardo

doi:10.1090/S0002-9947-2010-05283-3

Trace expansions for elliptic cone operators with stationary domains
HTML articles powered by AMS MathViewer

by Juan B. Gil, Thomas Krainer and Gerardo A. Mendoza PDF

Trans. Amer. Math. Soc. 362 (2010), 6495-6522 Request permission

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

Jochen Brüning and Robert Seeley, Regular singular asymptotics, Adv. in Math. 58 (1985), no. 2, 133–148. MR 814748, DOI 10.1016/0001-8708(85)90114-8
Jochen Brüning and Robert Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), no. 2, 369–429. MR 899656, DOI 10.1016/0022-1236(87)90073-5
J. Brüning and R. Seeley, The expansion of the resolvent near a singular stratum of conical type, J. Funct. Anal. 95 (1991), no. 2, 255–290. MR 1092127, DOI 10.1016/0022-1236(91)90030-9
Constantine J. Callias, The heat equation with singular coefficients. I. Operators of the form $-d^{2}/dx^{2}+\kappa /x^{2}$ in dimension $1$, Comm. Math. Phys. 88 (1983), no. 3, 357–385. MR 701923
Jeff Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR 530173, DOI 10.1073/pnas.76.5.2103
H. Falomir, M. A. Muschietti, and P. A. G. Pisani, On the resolvent and spectral functions of a second order differential operator with a regular singularity, J. Math. Phys. 45 (2004), no. 12, 4560–4577. MR 2105208, DOI 10.1063/1.1809257
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, DOI 10.1088/0305-4470/36/39/302
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), no. 26, 5427–5444. MR 1916056, DOI 10.1088/0305-4470/35/26/306
J. Gil, Heat trace asymptotics for cone differential operators, Ph.D. thesis, Universität Potsdam, 1998.
Juan B. Gil, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25–57. MR 1956600, DOI 10.1002/mana.200310020
Juan B. Gil, Thomas Krainer, and Gerardo A. Mendoza, Geometry and spectra of closed extensions of elliptic cone operators, Canad. J. Math. 59 (2007), no. 4, 742–794. MR 2338233, DOI 10.4153/CJM-2007-033-7
Juan B. Gil, Thomas Krainer, and Gerardo A. Mendoza, Resolvents of elliptic cone operators, J. Funct. Anal. 241 (2006), no. 1, 1–55. MR 2264246, DOI 10.1016/j.jfa.2006.07.010
Juan B. Gil, Thomas Krainer, and Gerardo A. Mendoza, On rays of minimal growth for elliptic cone operators, Modern trends in pseudo-differential operators, Oper. Theory Adv. Appl., vol. 172, Birkhäuser, Basel, 2007, pp. 33–50. MR 2308502, DOI 10.1007/978-3-7643-8116-5_{2}
—, Dynamics on Grassmannians and resolvents of cone operators, to appear in Analysis & PDE.
Juan B. Gil and Paul A. Loya, Resolvents of cone pseudodifferential operators, asymptotic expansions and applications, Math. Z. 259 (2008), no. 1, 65–95. MR 2375616, DOI 10.1007/s00209-007-0212-6
Juan B. Gil and Gerardo A. Mendoza, Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003), no. 2, 357–408. MR 1963689
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308
Gerd Grubb and Robert T. Seeley, Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Invent. Math. 121 (1995), no. 3, 481–529. MR 1353307, DOI 10.1007/BF01884310
Klaus Kirsten, Paul Loya, and Jinsung Park, The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^2/dr^2-1/(4r^2)$, J. Math. Phys. 47 (2006), no. 4, 043506, 27. MR 2226343, DOI 10.1063/1.2189194
Klaus Kirsten, Paul Loya, and Jinsung Park, 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, DOI 10.1007/s00229-007-0142-y
Klaus Kirsten, Paul Loya, and Jinsung Park, Exotic expansions and pathological properties of $\zeta$-functions on conic manifolds, J. Geom. Anal. 18 (2008), no. 3, 835–888. MR 2420767, DOI 10.1007/s12220-008-9028-9
M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Math. vol 136, B.G. Teubner, Stuttgart, Leipzig, 1997.
Paul Loya, On the resolvent of differential operators on conic manifolds, Comm. Anal. Geom. 10 (2002), no. 5, 877–934. MR 1957656, DOI 10.4310/CAG.2002.v10.n5.a1
Paul Loya, Parameter-dependent operators and resolvent expansions on conic manifolds, Illinois J. Math. 46 (2002), no. 4, 1035–1059. MR 1988248
Paul Loya, Patrick McDonald, and Jinsung Park, Zeta regularized determinants for conic manifolds, J. Funct. Anal. 242 (2007), no. 1, 195–229. MR 2274020, DOI 10.1016/j.jfa.2006.04.014
Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993. MR 1348401, DOI 10.1016/0377-0257(93)80040-i
Edith A. Mooers, Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999), 1–36. MR 1714065, DOI 10.1007/BF02791127
B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, Studies in Mathematics and its Applications, vol. 24, North-Holland Publishing Co., Amsterdam, 1991. MR 1142574
R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943