Computations of eigenvalues and resonances on perturbed hyperbolic surfaces with cusps (1812.05554v2)

Abstract: In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichm\"{u}ller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM.

• The paper introduces a method using the Neumann-to-Dirichlet map and finite elements to compute the scattering matrix on hyperbolic surfaces with cusps.
• The paper provides rigorous error estimates to ensure efficient and reliable computations of eigenvalues and resonances.
• The paper demonstrates its approach through detailed numerical studies on various configurations, including arithmetic surfaces and conformal perturbations.

Computational Methods for Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

The research article by Michael Levitin and Alexander Strohmaier presents an innovative approach to calculating eigenvalues and resonances on perturbed hyperbolic surfaces with cusps. Leveraging the Neumann-to-Dirichlet map and finite element methods, this work emphasizes a fast and direct computation of the scattering matrix, applicable to surfaces undergoing conformal perturbations or variations within the Teichmüller space.

The central theme revolves around the development of a methodological framework in which the Neumann-to-Dirichlet map—an operator relating Neumann boundary data to Dirichlet data on a compact manifold with the cusp removed—is instrumental in determining the scattering matrix. Despite the computational complexities typically associated with hyperbolic surfaces, this framework allows for efficient and accurate numerical computation. The authors illustrate that such computations, even when using finite element methods, can achieve notable accuracy.

Key Contributions and Findings

1. Numerical Computation of Scattering Matrix: The paper elucidates a method by which the scattering matrix can be effectively extracted from the Neumann-to-Dirichlet operator on hyperbolic surfaces with cusps. This matrix is a central object in scattering theory, encapsulating how incoming waves scatter from a surface.
2. Error Estimates: A significant part of this work is dedicated to providing error estimates for the scattering matrix, ensuring that the computational methods yield reliable results even when approximations are involved.
3. Resonances and Eigenvalues: The paper focuses on the numerical computation of resonances, which are poles of the meromorphic continuation of the resolvent of the hyperbolic Laplacian, as well as embedded eigenvalues. These numerical experiments are conducted for various hyperbolic structures, including arithmetic surfaces of genus one, with conformal perturbations explored to assess the robustness of the method.
4. Examples and Numerical Studies: Through detailed examples, the authors demonstrate the effectiveness of this approach across a spectrum of configurations. Special attention is given to arithmetic surfaces, where the scattering matrix is expressed in terms of known objects like the Riemann zeta function. The paper provides rich empirical data through graphs and tables, and even video animations showcasing the movement of scattering resonances.

Implications and Future Directions

The framework proposed has significant implications for both theoretical and computational mathematics, particularly in the context of spectral geometry and quantum chaos. The ability to compute eigenvalues and resonances in perturbed environments opens avenues for studying a broader class of geometric structures and could contribute to advancing our understanding of geometric scattering theory.

Furthermore, this methodology might inform developments in related fields such as mathematical physics, particularly in areas where wave behaviour in geometrically complex domains is of concern—ranging from quantum mechanics to acoustics. It also bridges connections with number theory, particularly through the explicit computation of scattering matrices on arithmetic surfaces, which often relate to deep questions regarding automorphic forms and spectral theory.

In summary, the work of Levitin and Strohmaier marks a substantial contribution to computational spectral theory, providing a robust toolset for exploring the intricate resonance behaviour on hyperbolic surfaces while ensuring the methods' adaptability through rigorous error control. Future research could look at extending these techniques to higher dimensions or more complex cusp configurations, as well as exploring applications to other types of partial differential equations on manifolds.