Isospectral

Abstract: Publisher Summary This chapter discusses three integrable hamiltonian systems connected with isospectral deformations. In recent years, various phenomena have been discovered that are clearly intimately related to integrable Hamiltonian systems yet they have very different origin. There are wave solutions of a nonlinear partial differential equation having a strong stability behavior. Originally these phenomena were brought to light by numerical experiments and later on related to the existence of infinitely many conservation laws that restrict the evolution of the solutions severely. As a Hamiltonian system in an infinite-dimensional function space, with a certain symplectic structure, and the conservation laws as integrals of this system, one can view this as an example of an integrable system of infinitely many degrees of freedom.

Abstract: Preface.-Chapter 1: Hyperbolic Structures.-Chapter 2: Trigonometry.- Chapter 3: Y-Pieces and Twist Parameters.- Chapter 4:The Collar Theorem.- Chapter 5: Bers' Constant and the Hairy Torus.- Chapter 6: The Teichmuller Space.- Chapter 7: The Spectrum of the Laplacian.- Chapter 8: Small Eigenvalues.- Chapter 9: Closed Geodesics and Huber's Theorem.- Chapter 10: Wolpert's Theorem.- Chapter 11: Sunada's Theorem.- Chapter 12: Examples of Isospectral Riemann surfaces.- Chapter 13: The Size of Isospectral Families.- Chapter 14: Perturbations of the Laplacian in Hilbert Space.-Appendix: Curves and Isotopies.-Bibliography.-Index.-Glossary.

Abstract: This paper introduces a method to extract 'Shape-DNA', a numerical fingerprint or signature, of any 2d or 3d manifold (surface or solid) by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach. Since the spectrum is an isometry invariant, it is independent of the object's representation including parametrization and spatial position. Additionally, the eigenvalues can be normalized so that uniform scaling factors for the geometric objects can be obtained easily. Therefore, checking if two objects are isometric needs no prior alignment (registration/localization) of the objects but only a comparison of their spectra. In this paper, we describe the computation of the spectra and their comparison for objects represented by NURBS or other parametrized surfaces (possibly glued to each other), polygonal meshes as well as solid polyhedra. Exploiting the isometry invariance of the Laplace-Beltrami operator we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary. Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart. Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues. This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum. With the help of this Shape-DNA, it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids. A patent application based on ideas presented in this paper is pending.

Abstract: A new approach to Hamiltonian structures of integrable systems is proposed by making use of a trace identity. For a variety of isospectral problems that can be unified to one model ψx=Uψ, it is shown that both the related hierarchy of evolution equations and the Hamiltonian structure can be obtained from the same solution of the equation Vx=[U,V].