Learn how to enable JavaScript on your browser. This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces.

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After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry closed geodesics and arithmetic prime numbers in proving the Selberg trace formula.

Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss.

The fruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results and then to be led towards very active areas in modern mathematics. His research interests are in geometry and automorphic forms, in particular the topology and spectral geometry of locally symmetric spaces. See All Customer Reviews. Shop Books. Add to Wishlist. USD Sign in to Purchase Instantly.

Overview This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. Table of Contents Preface.

## The Spectrum Of Hyperbolic Surfaces

Average Review. Write a Review. Properties of eigenvalues on Riemann surfaces with large symmetry groups Joseph Cook. References Publications referenced by this paper.

### Table of Contents

Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends David Borthwick. Recent developments in mathematical Quantum Chaos Steve Zelditch. Spectral theory of infinite-area hyperbolic surfaces David Borthwick. Fuchsian Groups. Uniform distribution of eigenfunctions on compact hyperbolic surfaces Steven Morris Zelditch. Integrating the second term in 5 by parts, and summing the equations 5 — 6 , we find that the spatial derivatives vanish. This energy loss through the boundaries is what the numerical method should mimic. Since boundary terms are negative semi-definite, we therefore have.

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In this section we define physical interface conditions that must be satisfied when elastic blocks are in contact. One idea of this paper is to use physical conditions to couple discontinuous Galerkin elements. Therefore, we focus here on a locked interface only, where slip motion is not permitted. However, the ideas expressed in this paper can be easily extended to other situations when internal slip is present. Since there are two characteristics going in and out of the interface we need exactly two interface conditions coupling the elastic subdomains.

We consider a simple linearized friction law. For later use, we summarize the interface condition:. However, we can model nonlinear frictional slip motion by replacing the second equation in 2. The elastic wave equation with the physical interface condition 2. Our main objective is to formulate an inter-element procedure using the physical interface condition 2.

The procedure should be formulated in a unified manner such that numerical flux functions are compatible with the general linear boundary condition 3 or 4. Extensions to higher space dimensions 2D and 3D should also be straightforward. We will now reformulate the boundary condition 3 and interface condition 2. The hat-variables will be constructed such that they preserve the amplitude of the outgoing characteristics and exactly satisfy the physical boundary conditions [ 3 ]. To begin, define the hat-variables preserving the amplitude of outgoing characteristics.

The algebraic problem for the hat-variables, defined by equations 13 and 15 , has a unique solution, namely.

The expressions in 3. It is particularly important to note that the boundary procedure 3. From 3. The first identity 18a holds by definition given in From the solutions of the hat-variables in 3. The algebraic identities 18a — 18c will be crucial in proving numerical stability. Equation 20 arises naturally in the boundary integral formulation of linear elasticity [ 37 ]. As before, combining both equations in 3. The above algebraic problem 23 has a unique solution which is solved exactly,.

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## Spectral Theory of Infinite-Area Hyperbolic Surfaces - David Borthwick - Google книги

The first identity 26a holds by the definition 3. The data is unique and exact. As before, the identities defined in 26a — 26c will be crucial in proving numerical stability.

Therefore, the weak form 5 — 6 yield. We will begin the development and construction of the inter-element and boundary procedure for the continuous weak form 27 — As we will see later the procedure and analysis will naturally carry over when numerical approximations are introduced. We will end the discussion with the derivation of an energy equation analogous to 8.

We construct flux fluctuations by penalizing data against incoming characteristics p and q ,. Thus, we have.

However, when numerical approximations are introduced numerical solutions can be discontinuous across element boundaries. Note also that the external physical boundary conditions and the inter-element conditions are treated in a unified manner. The penalty weights have been chosen such that the physical dimensions of all terms in equations 31 — 32 match.