Non-Scattering Energies and Transmission Eigenvalues in Hn

We consider non-scattering energies and transmission eigenvalues of compactly supported potentials in the hyperbolic spaces H-n. We prove that in H-2 a corner bounded by two hyperbolic lines intersecting at an angle smaller than 180 degrees always scatters, and that one of the lines may be replaced by a horocycle. In higher dimensions, we obtain similar results for corners bounded by hyperbolic hyperplanes intersecting each other pairwise orthogonally, and that one of the hyperplanes may be replaced by a horosphere. The corner scattering results are contrasted by proving discreteness and existence results for the related transmission eigenvalue problems.Peer reviewe

Uniqueness for the inverse boundary value problem with singular potentials in 2D

In this paper we consider the inverse boundary value problem for the Schrodinger equation with potential in Lp. We show that the potential is uniquely determined by the boundary measurements.Peer reviewe

Inverse problems for discrete heat equations and random walks

We study the inverse problem of determining a finite weighted graph $(X,E)$ from the source-to-solution map on a vertex subset $B\subset X$ for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on $B$. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$, or from the observation on $B$ of one realization of the random walk.Comment: 31 pages, 3 figures. arXiv admin note: text overlap with arXiv:2101.1002

Blockage detection in networks : The area reconstruction method

In this note we present a reconstructive algorithm for solving the cross-sectional pipe area from boundary measurements in a tree network with one inaccessible end. This is equivalent to reconstructing the first order perturbation to a wave equation on a quantum graph from boundary measurements at all network ends except one. The method presented here is based on a time reversal boundary control method originally presented by Sondhi and Gopinath for one dimensional problems and later by Oksanen to higher dimensional manifolds. The algorithm is local, so is applicable to complicated networks if we are interested only in a part isomorphic to a tree. Moreover the numerical implementation requires only one matrix inversion or least squares minimization per discretization point in the physical network. We present a theoretical solution existence proof, a step-by-step algorithm, and a numerical implementation applied to two numerical experiments.Peer reviewe

Unique Determination of the Shape of a Scattering Screen from a Passive Measurement

We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely, the obstacle is restricted to a two-dimensional plane and interacting with an arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant in the study and design of reflecting sonars and antennas, cases where one cannot assume that the incident wave is a plane wave. Our main result is that given the plane where the screen is located, the far-field pattern produced by any single arbitrary incident wave determines the exact shape of the screen, as long as it is not antisymmetric with respect to the plane. This holds even for screens whose shape is an arbitrary simply connected smooth domain. This is in contrast to earlier work where the incident wave had to be a plane wave, or more recent work where only polygonal scatterers are determined

Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions

This paper reports some interesting discoveries about the localization and geometrization phenomenon in plasmon resonances and the intrinsic geometric structures of Neumann-Poincaré eigenfunctions. It is known that plasmon resonance generically occurs in the quasi-static regime where the size of the plasmonic inclusion is sufficiently small compared to the wavelength. In this paper, we show that the global smallness condition on the plasmonic inclusion can be replaced by a local high-curvature condition, and the plasmon resonance occurs locally near the high-curvature point of the plasmonic inclusion. We link this phenomenon with the geometric structures of the Neumann-Poincaré (NP) eigenfunctions. The spectrum of the Neumann-Poincaré operator has received significant attentions in the literature. We show that the Neumann-Poincaré eigenfunctions possess some intrinsic geometric structures near the high-curvature points. We mainly rely on numerics to present our findings. For a particular case when the domain is an ellipse, we can provide the analytic results based on the explicit solutions

Gelfand's inverse problem for the graph Laplacian

We study the discrete Gelfand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: X = B U G, where B is called the "set of the boundary vertices" and G is called the "set of the interior vertices." We consider the case where the vertices in the set G and the edges connecting them are unknown. Assume that we are given the set B and the pairs symbolscript , symbolscript symbolscript where symbolscript are the eigenvalues of the graph Laplacian and symbolscript symbolscript are the values of the corresponding eigen-functions at the vertices in B. We show that the graph structure, namely the unknown vertices in G and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset S c G of cardinality ISI symbolscript 2 contains two extreme points. A point x E S is called an extreme point of S if there exists a point z E B such that x is the unique nearest point in S from z with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.Peer reviewe