Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

In this paper, we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.     

Quantum unique ergodicity

Consider a compact Riemannian manifold with ergodic geodesic flow. Quantum ergodicity is generalized from orthonormal bases of eigenfunctions of the Laplacian to packets of eigenfunctions. It is shown that this more general result is sharp. Namely, there may exist exceptional packets of eigenfunctions which concentrate on a submanifold.

     

Pseudo-Differential Calculus on Homogeneous Trees

To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L ² and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.     

The Cauchy process and the Steklov problem

Let X_t be a Cauchy process in . We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the “Mixed Steklov Problem.” Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (−1,1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric α-stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not available for more general symmetric α-stable processes.     

Concentration of Eigenfunctions Near a Concave Boundary

This article concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in different ways. We also prove one of these inequalities, which bounds the L ^p norms of the restrictions of eigenfunctions to broken geodesics.     

Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen (Sharp eigenvalue bounds and minimal surfaces in the ball, 2013). Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold (M, g) of dimension 2 into some k-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili (Int Math Res Not 17:939–952, 1999) which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface.     

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined‐field integral equation formulations for time‐harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the L² condition numbers for these formulations and also on the norms of the classical acoustic single‐ and double‐layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number k, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like k^1/3 as k →∞, when the scatterer is a circle or sphere, it can grow as fast as k^7/5 for a class of “trapping” obstacles. In this article, we prove further bounds, sharpening and extending our previous results. In particular, we show that there exist trapping obstacles for which the condition numbers grow as fast as exp(γk), for some γ > 0, as k →∞ through some sequence. This result depends on exponential localization bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low k. In the second part of the article, we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Calculus on the Sierpinski gasket I: polynomials, exponentials and power series

We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to xⁿ/n!. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.     

Ergodic semigroups of positivity preserving self-adjoint operators

We prove that an ergodic semigroup of positivity preserving self-adjoint operators is positivity improving. We also present a new proof (using Markov techniques) of the ergodicity of semigroups generated by spatially cutoff P(?)₂ Hamiltonians.     

Data analysis and representation on a general domain using eigenfunctions of Laplacian

We propose a new method to analyze and efficiently represent data recorded on a domain of general shape in by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the Karhunen–Loève transform/principal component analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.     

On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition

The Frankl problem without the spectral parameter was considered by Bitsadze and Smirnov. The present paper gives the eigenvalues and eigenfunctions of the Frankl problem with the odd parity condition. We prove the completeness of eigenfunctions. The Frankl problem with a nonlocal parity condition for the Lavrent’ev-Bitsadze equation is studied. The eigenvalues and eigenfunctions are found, and the basis property of the eigenfunctions in the elliptic part of the domain in the space L ₂ is proved.     

Patterson–Sullivan distributions in higher rank

For a compact locally symmetric space X _Γ of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an h-pseudo-differential calculus on X _Γ , we define and study lifted quantum limits as weak*-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson–Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.     

Robust Weak Ergodicity And Stable Ergodicity

In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r(r > 1) conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive (or negative) central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r(r > 1) conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.     

Harnack's inequality for a nonlinear eigenvalue problem on metric spaces

We prove Harnack's inequality for first eigenfunctions of the p-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (1,p)-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.     

Non-localization of eigenfunctions on large regular graphs

We give a delocalization estimate for eigenfunctions of the discrete Laplacian on large (d+1)-regular graphs, showing that any subset of the graph supporting ε of the L ² mass of an eigenfunction must be large. For graphs satisfying a mild girth-like condition, this bound will be exponential in the size of the graph.     

Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions

In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the Laplacian of the compact base in terms of its eigenfunctions. For those terms corresponding to zero modes, we obtain decay using analogs of estimates of Klainerman and Sideris. For the nonzero modes, estimates for Klein-Gordon equations, which provide better decay, are available.

     

The No Response Test for the reconstruction of polyhedral objects in electromagnetics

We develop a No Response Test for the reconstruction of a polyhedral obstacle from two or few time-harmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a response when the scatterer is inside of this test domain. We will prove that the No Response Test checks analytic continuability of a time-harmonic field from the far field pattern into the domain for a non-vibrating test domain B.We show that two incident waves, defined by one incident direction and two appropriately chosen directions of polarization, are enough to recover the convex hull of polyhedrals. Based on this uniqueness result, we build up the No Response Test and we prove convergence in the sense that it fully reconstructs a convex polyhedral scatterer D or the convex hull of an arbitrary polyhedral scatterer.Further, we will describe the algorithmic realization of the No Response Test and show the feasibility of the method by reconstruction of convex polyhedral objects in three dimensions. This is the first formulation of the No Response Test for electromagnetics.     

Spectral properties of a limit-periodic Schrödinger operator in dimension two

We study the Schrödinger operator H = ?Δ + V(x) in dimension two, V(x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis, and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves exp( $i\left\langle {\overrightarrow k ,\overrightarrow x } \right\rangle $ ) at the high energy region. Second, the isoenergetic curves in the space of momenta $\overrightarrow k $ corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.     

Complex zeros of real ergodic eigenfunctions

We determine the limit distribution (as λ→∞) of complex zeros for holomorphic continuations φ_λ ^? to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold (M,g) with ergodic geodesic flow. If \(\{\phi_{j_{k}}\}\) is an ergodic sequence of eigenfunctions, we prove the weak limit formula \(\frac{1}{\lambda_j}[Z_{\phi_{j_k}}^{\mathbb{C}}]\ \to\ \frac{i}{\pi} \partial\bar{\partial} |\xi|_g\), where \([Z_{\phi_{j_k}^{\mathbb{C}}}]\) is the current of integration over the complex zeros and where \(\overline{\partial}\) is with respect to the adapted complex structure of Lempert-Szöke and Guillemin-Stenzel.     

Trees as Brelot spaces

A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space which are called harmonic. We prove that by considering functions on a tree to be functions on the edges as well as on the vertices (instead of just on the vertices), a tree becomes a Brelot space. This leads to many results on the potential theory of trees. By restricting the functions just to the vertices, we obtain several new results on the potential theory of trees considered in the usual sense. We study trees whose nearest-neighbor transition probabilities are defined by both transient and recurrent random walks. Besides the usual case of harmonic functions on trees (the kernel of the Laplace operator), we also consider as “harmonic” the eigenfunctions of the Laplacian relative to a positive eigenvalue showing that these also yield a Brelot structure and creating new classes of functions for the study of potential theory on trees.