Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators

The operator $e^{-tA}$ and its trace are investigated in the case when $A$ is a non-self-adjoint elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter-ellipticity) we obtain a full asymptotic expansion in $t$ of the heat trace as $t\to 0^+$. As in the smooth compact case, the problem is reduced to the investigation of the resolvent $(A-\lambda)^{-1}$. The main step will consist in approximating this operator family by a parametrix to $A-\lambda$ using a suitable parameter-dependent calculus.Comment: 35 pages. Final version to appear in Math. Nachrichten. The paper has been improved. Section 4 has been rewritten and simplifie

Interrelation between radio and X-ray signatures of drifting subpulses in pulsars

We examined a model of partially screened gap region above the polar cap, in which the electron-positron plasma generated by sparking discharges coexists with thermionic flow ejected by the bombardment of the surface beneath these sparks. Our special interest was the polar cap heating rate and the subpulse drifting rate, both phenomena presumably associated with sparks operating at the polar cap. We investigated correlation between the intrinsic drift rate and polar cap heating rate and found that they are coupled to each other in such a way that the thermal X-ray luminosity $L_x$ from heated polar cap depends only on the observational tertiary subpulse drift periodicity $\hat{P}_3$ (polar cap carousel time). Within our model of partially screened gap we derived the simple formula relating $L_x$ and $\hat{P}_3$, and showed that it holds for PSRs B0943$+$10 and B1133+16, which are the only two pulsars in which both $L_x$ and $\hat{P}_3$ are presently known.Comment: 4 page

Hecke operators on rational functions

We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators, including the pleasing fact that the point spectrum of the operator U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet characters mod L, giving rise to the result that any arithmetic function of m that is completely multiplicative and also satisfies a linear recurrence must be a Dirichlet character times a power of m. We also define the notions of level and weight for rational eigenfunctions, by analogy with modular forms, and we show the existence of some interesting finite-dimensional subspaces of rational eigenfunctions (of fixed weight and level), whose union gives all of the rational functions whose coefficients are quasi-polynomials.Comment: 35 pages, LaTe

A conic manifold perspective of elliptic operators on graphs

We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators.Comment: 18 page

On cyclic numbers and an extension of Midy's theorem

In this note we consider fractions of the form 1/m and their floating-point representation in various arithmetic bases. For instance, what is 1/7 in base 2005? And, what about 1/4? We give a simple algorithm to answer these questions. In addition, we discuss an extension of Midy's theorem whose proof relies on elementary modular arithmetic.Comment: 6 pages, aimed at undergraduate student

Adjoints of elliptic cone operators

We study the adjointness problem for the closed extensions of a general b-elliptic operator A in x^{-\nu}Diff^m_b(M;E), \nu>0, initially defined as an unbounded operator A:C_c^\infty(M;E)\subset x^\mu L^2_b(M;E)\to x^\mu L^2_b(M;E), \mu \in \R. The case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator.Comment: 40 pages, LaTeX, preliminary versio

Resolvents of cone pseudodifferential operators, asymptotic expansions and applications

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.Comment: 30 pages, 5 figure

On the Noncommutative Residue and the Heat Trace Expansion on Conic Manifolds

Given a cone pseudodifferential operator $P$ we give a full asymptotic expansion as $t\to 0^+$ of the trace \Tr Pe^{-tA}, where $A$ is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains $\log t$ and new $(\log t)^2$ terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals.Comment: 15 page