Separation of variables in perturbed cylinders

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients. Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically yielding the eigenvalues of the operator. The same approach allows us to find complex resonances arising in some non-compact domains. We discuss numerical examples related to quantum waveguide problems.Comment: LaTeX 2e, 18 pages, 6 figure

Defining the spectral position of a Neumann domain

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. Morse-Smale complexes). This partition is generated by gradient flow lines of the eigenfunction -- these bound the so-called Neumann domains. We prove that the Neumann Laplacian $\Delta$ defined on a single Neumann domain is self-adjoint and possesses a purely discrete spectrum. In addition, we prove that the restriction of the eigenfunction to any one of its Neumann domains is an eigenfunction of $\Delta$. As a comparison, similar statements for a $nodal$ domain of an eigenfunction (with the Dirichlet Laplacian) are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, and hence is not necessarily continuous, so standard results about Sobolev spaces are not available

Spectral simplicity and asymptotic separation of variables

We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional. We also show that for all but countably many t, the geodesic triangle in the hyperbolic plane with interior angles 0, t, and t, has simple spectrum.Comment: 53 pages, 2 figure

Around quantum ergodicity

We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.Comment: 18 pages, 3 figures; minor revisions following the referee's comments. To appear special issue of Annales Math\'ematiques du Qu\'ebec in honor of Alexander Shnirelman's 75th birthda

Continuity properties of the inf-sup constant for the divergence

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary

The Steklov eigenvalue problem in a cuspidal domain

In this paper we analyze the approximation, by piecewise linear finite elements, of a Steklov eigenvalue problem in a plane domain with an external cusp. This problem is not covered by the literature and its analysis requires a special treatment. Indeed, we develop new trace theorems and we also obtain regularity results for the source counterpart. Moreover, under appropriate assumptions on the meshes, we present interpolation error estimates for functions in fractional Sobolev spaces. These estimates allow us to obtain appropriate convergence results of the source counterpart which, in the context of the theory of compact operator, are a fundamental tool in order to prove the convergence of the eigenpairs. At the end, we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests.Fil: Armentano, Maria Gabriela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Lombardi, Ariel Luis. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin