arXiv subjects

Clara L. Aldana

Publications and source records attributed to Clara L. Aldana.

10 recordsLinked to original sources

arXiv2026

On quasi-isospectrality of potentials and Riemannian manifolds

In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact manifolds and finite intervals, and then introduce the notion of quasi-isospectrality. We next investigate the BMT method as a systematic approach to constructing quasi-isospectral Sturm-Liouville operators on a finite interval, and apply it to several boundary value problems. Our main result shows that any two quasi-isospectral closed manifolds of odd dimension are, in fact, isospectral. In addition, we extend classical compactness results for isospectral potentials on low-dimensional manifolds to the quasi-isospectral setting via heat trace asymptotics.

math.SP↗

arXiv2024

The variation of Barnes and Bessel zeta functions

We consider the variation of two fundamental types of zeta functions that arise in the study of both physical and analytical problems in geometric settings involving conical singularities. These are the Barnes zeta functions and the Bessel zeta functions. Although the series used to define them do not converge at zero, using methods of complex analysis we are able to calculate the derivatives of these zeta functions at zero. These zeta functions depend critically on a certain parameter, and we calculate the variation of these derivatives with respect to the parameter. For integer values of the parameter, we obtain a new expression for the variation of the Barnes zeta function with respect to the parameter in terms of special functions. For the Bessel zeta functions, we obtain two different expressions for the variation via two independent methods. Of course, the expressions should be equal, and we verify this by demonstrating several identities for both special and elementary functions. We encountered these zeta functions while working with determinants of Laplace operators on cones and angular sectors.

math.NT↗

arXiv2023

Polyakov formulas for conical singularities in two dimensions

We investigate the zeta-regularized determinant and its variation in the presence of conical singularities, boundaries, and corners. For surfaces with isolated conical singularities which may also have one or more smooth boundary components, we demonstrate both a variational Polyakov formula as well as an integrated Polyakov formula for the conformal variation of the Riemannian metric with conformal factors which are smooth up to all singular points and boundary components. We demonstrate the analogous result for curvilinear polygonal domains in surfaces. We then specialize to finite circular sectors and cones and via two independent methods obtain variational Polyakov formulas for the dependence of the determinant on the opening angle. Notably, this requires the conformal factor to be logarithmically singular at the vertex. We further obtain explicit formulas for the determinant for finite circular sectors and cones.

math.SP↗

arXiv2019

Maximal determinants of Schrödinger operators on bounded intervals

We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\geq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $\wp$ function).

math.SP↗

arXiv2018

$A\_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gromov-Hausdorff topology. Our study is based on the use of $A_\infty$-weights from harmonic analysis, and provides geometric controls on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on $R^n$ with infinite volume and finite $L^{n/2}$ norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.

math.DG↗

arXiv2017

A Polyakov formula for sectors

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

math.SP↗

arXiv2013

Ricci flow and the determinant of the Laplacian on non-compact surfaces

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant curvature metric along which the determinant increases. The aim of this paper is to perform a similar analysis for the determinant of the Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic funnels or cusps. In that context, we show that the Ricci flow converges to a metric of constant curvature and that the determinant increases along this flow.

math.DG↗

arXiv2012

Asymptotics of relative heat traces and determinants on open surfaces of finite area

The goal of this paper is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps (M,g) and a metric h on the surface that is a conformal transformation of the initial metric g. We prove the existence of the relative determinant of the pair $(Δ_{h},Δ_{g})$ under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov's formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.

math.SP↗

arXiv2012

Compactness of relatively isospectral sets of surfaces via conformal surgeries

We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently studied by Borthwick and Perry, or to the closed case by Osgood, Phillips and Sarnak if there are only cusps.

math.DG↗

arXiv2011

Isoresonant conformal surfaces with cusps and boundedness of the relative determinant

We study the isoresonance problem on non-compact surfaces of finite area that are hyperbolic outside a compact set. Inverse resonance problems correspond to inverse spectral problems in the non-compact setting. We consider a conformal class of surfaces with hyperbolic cusps where the deformation takes place inside a fixed compact set. Inside this compactly supported conformal class we consider isoresonant metrics, i.e. metrics for which the set of resonances is the same, including multiplicities. We prove that sets of isoresonant metrics inside the conformal class are sequentially compact. We use relative determinants, splitting formulae for determinants and the result of B. Osgood, R. Phillips and P. Sarnak about compactness of sets of isospectral metrics on closed surfaces. In the second part, we study the relative determinant of the Laplace operator on a hyperbolic surface as function on the moduli space. We consider the moduli space of hyperbolic surfaces of fixed genus and fixed number of cusps. We consider the relative determinant of the Laplace operator and a model operator defined on the cusps. We prove that the relative determinant tends to zero as one approaches the boundary of the moduli space.

math.SP↗