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Register a new userOn the resolvent and spectral functions of a second order differential operator with a regular singularity
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We consider the resolvent of a second order differential operator with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents unusual powers of $lambda$ which depend on the singularity. The consequences for the pole structure of the $zeta$-function, and the small-$t$ asymptotic expansion of the heat-kernel, are also discussed.
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We consider the resolvent of a system of first order differential operators with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents powers of $lambda$ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding $zeta$ and $eta$-functions are also discussed.
One of the many problems to which J.S. Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-$t$ asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities, as the Casimir energy. In this article we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of $t$, and even negative integer powers of $log{t}$, in this asymptotic expansion for the selfadjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the $zeta$-function associated to these selfadjoint extensions presents an unusual analytic structure.
We get a generalization of Kreins formula -which relates the resolvents of different selfadjoint extensions of a differential operator with regular coefficients- to the non-regular case $A=-partial_x^2+( u^2-1/4)/x^2+V(x)$, where $0< u<1$ and $V(x)$ is an analytic function of $xinmathbb{R}^+$ bounded from below. We show that the trace of the heat-kernel $e^{-tA}$ admits a non-standard small-t asymptotic expansion which contains, in general, integer powers of $t^ u$. In particular, these powers are present for those selfadjoint extensions of $A$ which are characterized by boundary conditions that break the local formal scale invariance at the singularity.
We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrodinger operators on graphs.
The zero locus of a bivariate polynomial $P(x,y)=0$ defines a compact Riemann surface $Sigma$. The fundamental second kind differential is a symmetric $1otimes 1$ form on $Sigmatimes Sigma$ that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newtons polygon of $P$, involving only integer combinations of products of coefficients of $P$. Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of $P$.
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