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Register a new userA transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions
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Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $Gammabackslashmathbb{H}$ with cusps and all finite-dimensional unitary representations $chi$ of $Gamma$. The eigenfunctions with eigenvalue $1$ of the fast transfer operators determine the zeros of the Selberg zeta function for $(Gamma,chi)$. Further, if $Gamma$ is cofinite and $chi$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue $1$ of the slow transfer operators characterize Maass cusp forms for $Gamma$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $Gammabackslashmathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $chi$ of the Hecke triangle group $Gamma$. In particular we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Moller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.
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We initiate the study of Selberg zeta functions $Z_{Gamma,chi}$ for geometrically finite Fuchsian groups $Gamma$ and finite-dimensional representations $chi$ with non-expanding cusp monodromy. We show that for all choices of $(Gamma,chi)$, the Selberg zeta function $Z_{Gamma,chi}$ converges on some half-plane in $mathbb{C}$. In addition, under the assumption that $Gamma$ admits a strict transfer operator approach, we show that $Z_{Gamma,chi}$ extends meromorphically to all of $mathbb{C}$.
We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmuller space. We then use this formula to determine the asymptotic behavior as $text{Re} (s) to infty$ of the second variation. As a consequence, for $m in mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, mathcal K_t)to tin mathcal T$ of holomorphic m-differentials over the Teichmuller space $mathcal T$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.
We study elements of the spectral theory of compact hyperbolic orbifolds $Gamma backslash mathbb{H}^{n}$. We establish a version of the Selberg trace formula for non-unitary representations of $Gamma$ and prove that the associated Selberg zeta function admits a meromorphic continuation to $mathbb{C}$.
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $zeta$-functions to efficiently compute values of spectral $zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $tau$. Depending on the underlying boundary conditions, we express the $zeta$-function values in terms of a fundamental system of solutions of $tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $zeta$-function through a Liouville transformation and provide an explicit expression for the $zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schr{o}dinger operators with zero, piecewise constant, and a linear potential on a compact interval.
In an earlier paper (A. N. Kochubei, {it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirovs fractional differentiation operator $D^alpha$, $alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^alpha$ that the appropriate change of variables reduces equations with $D^alpha$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we begin an operator-theoretic investigation of the operator $I^alpha$, and study a related analog of the Laplace transform.
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