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2016 | Buch

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Spectral Theory of Infinite-Area Hyperbolic Surfaces

verfasst von: David Borthwick

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added.

Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution.

The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields.

Review of the first edition:

"The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

A hyperbolic surface is a surface with geometry modeled on the hyperbolic plane. Spectral theory in this context refers generally to the Laplacian operator induced by the hyperbolic structure. Selberg [244] pioneered the study of spectral theory of hyperbolic surfaces in the 1950s, drawing inspiration from earlier work of Maass [162]. Motivated by analogies to the classical zeta and theta functions of number theory, Selberg applied tools and ideas from spectral theory and harmonic analysis to the study of automorphic forms associated with Fuchsian groups. This led in particular to beautiful formulas connecting the geometry of compact hyperbolic surfaces to the spectral theory.

David Borthwick

Chapter 2. Hyperbolic Surfaces

Abstract

For the purposes of this book, a surface is a connected, orientable two-dimensional smooth manifold, without boundary unless otherwise specified. Throughout the book we will restrict our attention to surfaces which are topologically finite, meaning that the surface is homeomorphic to a compact surface with finitely many points excised. An end is an equivalence class of neighborhoods which are contractible to one of these excised points. Topologically finite surfaces are classified up to diffeomorphism by the genus g and the number of ends n. The corresponding value of the Euler characteristic is \(\chi = 2 - 2g - n\). An example is shown in Figure 2.1.

David Borthwick

Chapter 3. Selberg Theory for Finite-Area Hyperbolic Surfaces

Abstract

To set the stage for the development of the spectral theory in the infinite-area case, we will first review some details of the spectral theory for compact and finite-area hyperbolic surfaces (Fuchsian groups of the first kind).

David Borthwick

Chapter 4. Spectral Theory for the Hyperbolic Plane

Abstract

In our discussion of spectral theory we naturally start with the hyperbolic plane itself, the primary example of a hyperbolic surface of infinite area.

David Borthwick

Chapter 5. Model Resolvents for Cylinders

Abstract

In this chapter we’ll develop explicit formulas for the resolvent kernels of the other elementary surfaces: the hyperbolic and parabolic cylinders. These explicit formulas will serve as building blocks when we turn to the construction of the resolvent in the general case in Chapter 6 This is because of the decomposition result of Theorem 2.23, which showed that the ends of non-elementary hyperbolic surfaces are funnels and cusps.

David Borthwick

Chapter 6. The Resolvent

Abstract

In Chapters 4 and 5 we worked out the resolvent kernels for the elementary surfaces. This provides a set of model resolvents for funnels and cusps in particular, which are the only possible end types in a non-elementary geometrically finite hyperbolic surface by Theorem 2.23.

David Borthwick

Chapter 7. Spectral and Scattering Theory

Abstract

The basic spectral theory of the Laplacian on a geometrically finite hyperbolic manifold was worked out by Lax-Phillips [148‐151], in the abstract framework of Lax-Phillips scattering theory [152].

David Borthwick

Chapter 8. Resonances and Scattering Poles

Abstract

The physical concept of resonance refers to the disproportionate response of an oscillating system driven by an external force at a frequency that is close to one of its natural frequencies of vibration. For example, an external tone will cause a strong sympathetic vibration in a violin string if the frequency matches the tuning of the string.

David Borthwick

Chapter 9. Growth Estimates and Resonance Bounds

Abstract

We introduced the resonance set \(\mathcal{R}_{X}\) for a surface with hyperbolic ends in Chapter 8 Resonances are assumed to be repeated in \(\mathcal{R}_{X}\) according to the multiplicity m(ζ) defined in (8.4).

David Borthwick

Chapter 10. Selberg Zeta Function

Abstract

For a geometrically finite hyperbolic surface X the Selberg zeta function Z _X(s) was introduced in §2.5. The zeta function is associated with the length spectrum of X (or, equivalently, to traces of conjugacy classes of Γ). We will see in this chapter that it deserves to be thought of as a spectral invariant as well, by virtue of a beautiful correspondence between resonances of X and the zeros of Z _X(s).

David Borthwick

Chapter 11. Wave Trace and Poisson Formula

Abstract

On a compact manifold, the wave trace is defined to be the distributional trace of the wave operator, \(U(t):= e^{it\sqrt{\Delta }}\). This is easily seen to be a spectral invariant, because it can be written explicitly as

David Borthwick

Chapter 12. Resonance Asymptotics

Abstract

One implication of the trace formula of Corollary 11.5 is immediately clear: for a non-elementary geometrically finite hyperbolic surface the set \(\mathcal{R}_{X}\) must be infinite, to account for the singularities on the right-hand side of (11.20). The trace formula contains much more information on the distribution of resonances, and in this chapter we will discuss methods by which this information can be extracted.

David Borthwick

Chapter 13. Inverse Spectral Geometry

Abstract

Determining the spectral properties of the Laplacian on a given Riemannian manifold is a “forward” spectral problem. The corresponding “inverse” problem is to deduce geometric properties from some knowledge of the spectrum. In the case of a surface with hyperbolic ends, the input data could include the resonance set, the scattering phase, perhaps even the scattering matrix for a particular set of frequencies.

David Borthwick

Chapter 14. Patterson-Sullivan Theory

Abstract

The exponent of convergence δ of a Fuchsian group Γ was defined in (2.20). We also noted some basic facts about the exponent:

David Borthwick

Chapter 15. Dynamical Approach to the Zeta Function

Abstract

The definition (2.23) of Selberg’s zeta function as a product over the length spectrum was in some sense very convenient, because of the link between conjugacy classes of Γ and closed geodesics furnished by Proposition 2.25. On the other hand, existence of a meromorphic continuation was far from obvious and controlling the growth of the zeta function was quite difficult (see the proof of Theorem 10.1). There is an alternative framework for zeta functions which is much more general than Selberg’s original definition, and basic properties such as meromorphic continuation are much easier to prove from this point of view. The essential idea is to associate the zeta function to the dynamics of the geodesic flow on the surface (rather than to the geometry of the surface). This viewpoint leads to a definition which can be applied to any dynamical system under certain conditions on the flow. The main drawback to this approach, in the context of hyperbolic surfaces, is the fact that these methods do not apply when the surface has cusps because of the non-compactness of the convex core. In this section we restrict our attention to geometrically finite hyperbolic surfaces without cusps (convex cocompact Fuchsian groups).

David Borthwick

Chapter 16. Numerical Computations

Abstract

Because resonances are defined by meromorphic continuation, direct computation via the resolvent is very difficult. The product definition (2.23) of the Selberg zeta function has the same difficulty; the formula does not apply in the region of interest. However, for hyperbolic surfaces without cusps, the dynamical zeta function introduced in §15.3 provides a suitable alternative. The transfer operator is trace-class for any value of \(s \in \mathbb{C}\), so analytic continuation is not required.

David Borthwick

Backmatter

Titel: Spectral Theory of Infinite-Area Hyperbolic Surfaces
verfasst von: David Borthwick
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-33877-4
Print ISBN: 978-3-319-33875-0
DOI: https://doi.org/10.1007/978-3-319-33877-4