Exotic Expansions and Pathological Properties of Ζ-functions on Conic Manifolds
نویسندگان
چکیده
We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general selfadjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.
منابع مشابه
The ubiquitous ζ-function and some of its “usual” and “unusual” meromorphic properties
In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of ζ-functions associated to conic manifolds proved in [37]. In particular, we show that the meromorphic extensions of these ζfunctions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multipli...
متن کاملCoefficient Estimates for Some Subclasses of Analytic and Bi-Univalent Functions Associated with Conic Domain
The main objective of this investigation is to introduce certain new subclasses of the class $Sigma $ of bi-univalent functions by using concept of conic domain. Furthermore, we find non-sharp estimates on the first two Taylor-Maclaurin coefficients $ left vert a_{2}right vert $ and $left vert a_{3}right vert $ for functions in these new subclasses. We consider various corollaries and consequen...
متن کاملEigenfunction Expansions for Second-Order Boundary Value Problems with Separated Boundary Conditions
In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....
متن کاملA Resolvent Approach to Traces and Zeta Laurent Expansions
Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TRA introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C0(A, P ) in the trace expansion of A(P−λ) (with an auxiliary elliptic operator P ), as determined in a joint work with Seeley 1995. The definiti...
متن کاملCalabi-Yau Manifolds Over Finite Fields, II
We study ζ-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The ζ-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to ‘see’ these curves in the geometry of the quin...
متن کامل