The spectral theory of compact and of finite area Riemannian surfaces has a long history and is relatively well understood. However, the case of infinite area hyperbolic surfaces has been understood only relatively recently and the theory differs in many important regards from the finite case. The resolvent of the Laplacian plays an important role – scattering theory and resonances play a role similar to that played by discrete eigenvalues in the finite theory. The author deals with the context of hyperbolic surfaces although many of the results could be stated more generally – this enables the author to present a relatively self-contained treatment of the subject.
Chapter 1 contains a brief introduction to the subject. Chapter 2 treats hyperbolic surfaces (the hyperbolic plane, Fuchsian groups, geometrically finite groups, classification of hyperbolic ends, Gauss-Bonnet theorem, length spectrum, Selberg zeta function) and Chapter 3 treats compact and finite area surfaces (Selberg’s trace formula, consequences of the trace formula, finite area hyperbolic surfaces). Spectral theory (resolvent, generalized eigenfunctions, scattering matrix), model resolvents for cylinders (hyperbolic cylinders, funnels, parabolic cylinder), the resolvent (compactification, analytic Fredholm theorem, continuation of the resolvent, structure of the resolvent kernel, the stretched product), and spectral and scattering theory (essential and discrete spectrum, absence of embedded eigenvalues, generalized eigenfunctions, scattering matrix, scattering matrices for the funnel and cylinders) are treated in Chapters 4–7.
Chapters 8–10 deal with resonances and scattering poles (multiplicities of resonances, structure of the resolvent at a resonance, scattering poles, operator logarithmic residues, half-integer points, coincidence of resonances, scattering poles), with upper bound for resonances (resonances and zeros of determinants, singular value estimates, estimates on model terms), and the Selberg zeta function (relative scattering determinant, regularized traces, the resolvent 0-trace calculation, structure of the zeta function, order bound, determinant of the Laplacian).
Chapters 11–13 deal with wave trace and Poisson formula (regularized wave trace, model wave kernel, wave 0-trace formula, Poisson formula), with resonance asymptotics (lower bound on resonances, lower bound near the critical line, Weyl formula for the scattering phase), and with inverse spectral geometry (resonances and the length spectrum, hyperbolic trigonometry, Teichmüller space, finiteness of isospectral classes).
Patterson-Sullivan theory (ergodicity, Hausdorff measure of the limit set, the first resonance, prime geodesic theorem, refined asymptotics of the length spectrum) and the dynamical approach to the zeta function (Schottky groups, symbolic dynamics, dynamical zeta function, growth estimates) are dealt with in the final two chapters. The book concludes with an appendix (entire functions, distributions and Fourier transforms, spectral theory, singular values, traces, and determinants, and pseudo-differential operators).
The book is suitable both for experts in the field and also for graduate students with a basic knowledge of algebra, differential geometry, and topology and also a bit of advanced knowledge concerning analysis. The 228 bibliographic entries strike a useful balance between providing the basic references in the field and trying to summarize (an impossible task) all of spectral geometry.