Quasinormal frequencies are the damped modes of oscillation associated with linearized perturbations of a black hole. In this thesis we study the distribution of these frequencies in the complex plane for a class of solutions to the Einstein equations with negative cosmological constant; these are the so-called anti-de Sitter spacetimes. Particular emphasis is placed on the family of rotating Kerr--AdS black holes.

Quantum decay rates appear as imaginary parts of resonances, or poles of the meromorphic continuation of the resolvent of the Laplacian. As energy grows, decay rates are related to properties of geodesic flow and to the structure at infinity. In this thesis we study resonances for manifolds with infinity given by hyperbolic funnels, and establish relationships between trapping of geodesic trajectories and distribution and density of resonances.

We find the largest resonance free regions, corresponding to the fastest decay, in the nontrapping case, that is the case in which all geodesics escape to infinity. When the manifold is trapping, but the trapping is hyperbolic and sufficiently mild, we find resonance free strips with the width of the strip given by the topological pressure. For manifolds with hyperbolic dynamics, but possibly more substantial trapping, we find upper bounds on the rate of accumulation of resonances to the essential spectrum.

We use the method of complex scaling to holomorphically deform our operator and to realize the resonances as eigenvalues of the deformed, nonselfadjoint operator, which allows us to study them directly using microlocal estimates. The principal contribution of this thesis is the adaptation of the method of complex scaling to manifolds with hyperbolic ends, and the ``gluing'' of the estimates it provides near infinity to estimates near the trapped set which we take from other authors.

Berezin--Toeplitz operators are quantizations of functions on K\"ahler manifolds equipped with a positive line bundle $L$. When the K\"ahler manifold $M$ is compact, the quantization procedure associates every smooth function $f\in C^\infty(M)$ to a family of matrices $T_N f $ indexed by $N\in \N$, whose size goes to infinity as $N\to \infty$ (corresponding to the semiclassical limit $h \to 0$). Each matrix $T_N f$ acts on the finite-dimensional space of holomorphic sections of the $N$th tensor power of $L$.

In this thesis we study the spectrum of $T_N f + \delta \mathcal{G}_\omega(N)$ where $\delta = \delta (N) > 0$ and $\mathcal{G}_\omega(N)$ is a family of random matrices. Under certain conditions, the spectrum satisfies a probabilistic Weyl law involving $f$ and the volume form on the K\"ahler manifold. Specifically, as $N\to \infty$, the (normalized) empirical spectral distribution of $T_N f + \delta \mathcal{G}_\omega(N)$ converges weakly almost surely to the (normalized) push-forward by $f$ of the Liouville volume form on $M$. This generalizes a result of Martin Vogel, which considered the case of torii.

Proving this result requires extending the usual calculus of Berezin--Toeplitz operators to an exotic class of functions.The exotic nature of these functions (classical observables) refers to the property that their derivatives are allowed to grow in ways controlled by local geometry and the power of the line bundle. The properties of this quantization are obtained via careful analysis of the kernels of the operators using Melin and Sj\"ostrand’s method of complex stationary phase. For this more exotic class of functions, we obtain a functional calculus result, a trace formula, and a parametrix construction.

This thesis explores the spectral properties of Schrödinger-type operators on various

domains such as R^2, rectangles, and metric graphs. In particular, we consider special types of operators called point scatterers that act as the Laplacian away from a discrete set of points. Such a model provides a simple tool to study how the presence of point-wise potentials perturbs the spectral properties of the Laplacian.

In Chapter 1, we introduce the general procedure to properly define the point scatterers

on a general domain. The theory of self-adjoint extension and Krein’s formula play important

roles in the process.

In Chapter 2, we formulate point scatterers in R^2 using the renormalization process

mentioned in the previous chapter. We start with the one-point scatterer which is the

simplest case and then generalize the result to finitely many point scatterers and infinitely

many point scatterers. Then we consider a special case in which the scatterers are placed

periodically as a combination of infinitely many point scatterers and the Floquet-Bloch

theory of solid-state physics for crystal structures. As an application inspired by carbon

nano-structures such as graphene, we prove that honeycomb lattice point scatterers generate

conic singularities on the dispersion relation.

In Chapter 3, we consider a point scatterer on a rectangular domain to investigate how

the eigenfunctions on the rectangle are affected by the point-wise perturbation. We prove

that a point scatterer eventually acts as a barrier confining the eigenfunction as the domain

gets thinner.

In Chapter 4, we introduce how the point scatterers can be incorporated with the notion of

quantum graphs. In addition, the resonances of quantum graphs are investigated. We provide

the quantum graph version of a Fermi golden rule, which provides an explicit expression for

the infinitesimal change of states in terms of the scattering resonances.

In the setting of obstacle scattering in Euclidean spaces, the poles of meromorphic continuation of the resolvent of the Laplacian on the exterior region are called the resonances or scattering poles. Each resonance corresponds to a resonant wave. The real part of a resonance corresponds to the frequency of the wave, while the imaginary part corresponds to the decay rate of the wave. Consequently understanding the distribution of the resonances is important in understanding the long time behavior of the solution to wave equations in the exterior domain.

We study the distribution of resonances in the case of a strictly convex obstacle with smooth boundary. In particular, under general boundary conditions, we prove the existence of the cubic resonance free regions near the real axis. Moreover, if the obstacle is close to a sphere, in the sense that it satisfies certain pinched curvature conditions, we prove that the resonances close to the real axis are separated into cubic bands and in each band, the counting function of resonances satisfies a Weyl law. We also generalize these results to totally convex obstacles in more general asymptotic Euclidean metrics.

Scattering resonances are the analogues of eigenvalues for problems on non-compact domains. The real part and imaginary part of the resonances capture the rates of oscillation and decay of the scattering waves. Hence the location of resonances reflects the long-time behavior of the waves on non-compact domains.

In this thesis we study a computational technique for scattering resonances, that is the method of \emph{complex absorbing potentials} (CAP). We show that the CAP method for computing resonances applies to the case of scattering by exponentially decaying potentials. We also show that the CAP method is valid for an abstractly defined class of \emph{black box} perturbations of dilation analytic second order differential operators which is close to the Laplacian near infinity. The black box formalism allows a unifyingtreatment of diverse problems ranging from obstacle scattering to scattering on finite volume surfaces without addressing the details of specific situations.

The black box scattering problem motivates us to study the boundary perturbations in obstacle scattering. We show that all resonances in obstacle scattering with Dirichlet boundary condition are generically simple in the class of obstacles with $C^k$ (and $C^\infty$) boundaries, $k \geq 2$. This generalizes the case of eigenvalues of second order elliptic operators on a compact domain that all eigenvalues are simple for a generic compact domain.

We investigate the stability of complex numbers called resonances in certain open chaotic systems. In the context of waves scattered by a potential, scattering resonances are complex numbers that quantize exponential decay rates of the local energy. In the context of

hyperbolic dynamical systems, Pollicott--Ruelle resonances quantize exponential decay rates of statistical correlations. We will show that resonances occurring in these two situations are stochastically stable. This theoretical result supports the possibility of observing resonances in experimental physics.

This dissertation consists of two independent chapters, based on two papers of the author.

Chapter 1 focuses on scattering resonances. We give a simple model for waves propagating through a localized disordered crystal with small typical scale of heterogeneity. Roughly speaking, our results show that resonances (hence propagating waves) are only weakly perturbed by the crystal. This chapter is organized as follows:

- We describe our model and state our theorems in S1.1. We relate these results to previous study in the idealized case of deterministic highly oscillatory perturbations, whose understanding is very important for the proofs.

- In S1.2, we give an overview of the theory of scattering resonances. We define them using the stationary Schrodinger resolvent and we relate them to local energy decay of waves. This makes Chapter 1 essentially self-contained.

- S1.3 studies the general perturbation theory of scattering resonances. We work with a parameter that typically quantifies the degree of oscillations of a random potential. When this parameter is small enough, we characterize locally scattering resonances as

the zero set of a holomorphic function.

- In S1.4, we give a version of the Hanson--Wright inequality. This is a large deviation estimate for a quadratic form evaluated at random vectors with many entries. It is crucial in the rest of the proof. We also derive a modified version of the central limit

theorem, based on Lindeberg's theorem.

- S1.5 is the core of the chapter. The tools of S1.4 show that the general theory developed in x1.3 applies to random highly oscillatory potentials. In particular, we can write their resonances as the zeroes of a random holomorphic function. We show that this random

function can (roughly speaking) be written as a rescaled Gaussian -- modulo negligeable terms. The variance of the Gaussian arises from large deviation effects, while the average of the Gaussian comes from constructive interference among oscillatory terms

(an effect that was thoroughly studied in the context of deterministic highly oscillatory potentials).

- In S1.6, we prove the stochastic stability theorems. These are valid with high probability. We conclude by exhibiting an example (that appears with small probability) where the conclusions of the theorems do not hold.

Chapter 2 is a study of hyperbolic dynamical systems perturbed by a white noise. These form stochastic processes called kinetic Brownian motion. We use a recent microlocal approach to define Pollicott--Ruelle resonances via kinetic Brownian motion. This is a form of stochastic stability. The presentation is as follows:

- In S2.1, we state our stochastic stability result: the eigenvalues of the generator of the kinetic Brownian motion approach Pollicott--Ruelle resonances in the small white noise regime. We give an overview of recent results for random perturbations of the geodesic equation, and we describe related facts about the hypoelliptic Laplacian of Bismut.

- In S2.2, we recall a modern perspective on Pollicott--Ruelle resonances. It consists of seeing them as spectral quantities rather than dynamical one. This requires the semiclassical construction of anisotropic Sobolev spaces, which improve regularity in the contracting direction of the flow and lower it in the expanding direction. We give an axiomatic introduction to microlocal and semiclassical analysis.

- S2.3 presents the kinetic Brownian motion as a perturbation of the geodesic equation. We also describe its lift to the orthonormal frame bundle (a step required in S2.4).

- In S2.4, we prove a subelliptic estimate for the generator of the kinetic Brownian motion. An informal probabilistic statement is as follows: the white noise perturbation is not too large compared to the kinetic Brownian motion itself.

- S2.5 reformulates the subelliptic estimate in the context of anisotropic Sobolev spaces that are needed to define Pollicott--Ruelle resonances.

- In S2.6, we show that the subelliptic estimate allows to control high frequencies of the white noise perturbation; and that the low frequencies of the white noise perturbations can be treated as an absorbing potential. This enables us to prove the main theorem.

Zeroth order pseudodifferential operators with Morse--Smale dynamical assumption have been used as a model for the dynamics of internal waves. The operators provide a novel mathematical explanation to the formation of attractors for two dimensional internal waves in a bounded aquarium. In addition, the spectral properties are somewhat surprisingly related to phenomena in quantum scattering.

In this thesis we introduce an analogue of the scattering matrix and describe its geometric structure. We also study the dynamics of resonances and answer a question posed by Colin de Verdi ere by proving the generic absence of embedded eigenvalues.

The definition of the scattering matrix, although based on analogues with Euclidean scattering, is more involved as we need to deal with the principal symbols of Lagrangian distributions. After being conjugated by certain natural unitary operators, our scattering matrix turns out to be a Fourier integral operator and its canonical relation is related to the bicharacteristics of the operator.

Resonances for 0th order pseudodifferential operators are defined as limits of eigenvalues for the operator with viscosity. We show that the convergences rate is linear for simple embedded eigenvalues of the operator. We also consider 0th order perturbations of 0th pseudodifferential operators. A series expansion of resonances for the perturbed operators near embedded eigenvalues for the original operator is proved. A special case of this expansion gives the Fermi golden rule. Using the formulae for the coefficients in the expansion, we prove the absence of embedded eigenvalues for generic perturbation.

As a by-product of the study of 0th order pseudodifferential operators, we state and prove the sharp radial estimates in Besov spaces, which generalize the corresponding estimates in Sobolev spaces.

We study decay properties of linear waves on Kerr and Kerr-de Sitter black hole backgrounds. We are particularly interested in quasi-normal modes (QNMs), which are the complex frequencies of exponentially decaying and oscillating solutions to the wave equation. For slowly rotating Kerr-de Sitter black holes, we show that QNMs obey a quantization condition; that is, they lie asymptotically on a lattice. We also obtain a resonance expansion of linear waves in terms of QNMs. For the general Kerr-de Sitter case and its small stationary perturbations, we use r-normal hyperbolicity of the set of trapped lightlike rays to obtain a band of QNMs with a Weyl law.

Semiclassical analysis and other types of asymptotic analysis are important tools in the study of partial differential equations. We demonstrate the versatility of these methods by proving two types of results.

First we study the dynamics of solitary waves for two different nonlinear Schr\"odinger equations. The first is the Hartree equation the second is a general nonlinear Schr\"odinger equation with an $L^2$-subcritical power nonlinearity. For both of these equations, we show that a solution found using initial conditions within $\eps \le \sqrt h$ of a soliton (in $H^1$) evolves according to the equations of motion given by the effective Hamitonian up to time $\sim |\log h|/h$, with errors up to size $\eps +h^2$. We achieve this result using the methods of Holmer-Zworski and the spectral results of Lenzmann and Weinstein.

Next we study the spectral properties of $n$-dimensional semiclassical radial Schr\"odinger operators. We prove that their spectrum determines the potential within a large class of potentials for which we assume no symmetry or analyticity. Our proof is analogous to the spectral determination of the sphere and uses the first two semiclassical trace invariants and the isoperimetric inequality.